Table Of ContentObservation and calculation of the quasi-bound rovibrational levels of the electronic
ground state of H+
2
Maximilian Beyer and Fr´ed´eric Merkt
Laboratorium fu¨r Physikalische Chemie, ETH Zu¨rich, 8093 Zu¨rich, Switzerland
(Dated: January 16, 2017)
Although the existence of quasi-bound rotational levels of the X+ 2Σ+ ground state of H+ has
g 2
been predicted a long time ago, these states have never been observed. Calculated positions and
widths of quasi-bound rotational levels located close to the top of the centrifugal barriers have not
beenreportedeither. GiventherolethatsuchstatesplayintherecombinationofH(1s)andH+ to
formH+,thislackofdatamayberegardedasoneofthelargestunknownaspectsofthisotherwise
7 2
accurately known fundamental molecular cation. We present measurements of the positions and
1
widthsofthelowest-lyingquasi-boundrotationallevelsofH+ andcomparetheexperimentalresults
0 2
with the positions and widths we calculate using a potential model for the X+ state of H+ which
2 2
includes adiabatic, nonadiabatic, relativistic and radiative corrections to the Born-Oppenheimer
n approximation.
a
J
3 ThetheoreticaltreatmentofH+ hasplayedandisstill nances located above the H(1s) + H+ dissociation limit,
2
1
playinganimportantroleinthedevelopmentofquantum butbelowthemaximaoftherelevantcentrifugalbarriers
chemistry. H+ possessesoneelectronanditsenergy-level [12]. Whereas 26 of these resonances are extremely nar-
] 2
h structure can be calculated with extraordinary accuracy. row and can be calculated as accurately as bound levels,
p Highly accurate ab-initio calculations started with the Mosslists13quasi-boundlevelsforwhichanaccuracyof
-
m nonadiabatic theory of Kolos and Wolniewicz [1] and its 10−4cm−1couldnotbereached. 19quasi-boundlevelsof
applications to H [2–6] and H+ [7–10]. To compute theX+stateareevenlocatedsoclosetothetopofthere-
e 2 2
h nonadiabaticeffectsinH+2, approachesbasedonatrans- spectivecentrifugalpotentialbarriersthattheycouldnot
c formed Hamiltonian [11, 12] and coordinate-dependent be calculated so far [12, 16]. These levels have not been
s. vibrationalandrotationalmasses[13–17]areparticularly observedexperimentallyeither. Thislackofknowledgeis
c successful. Promising alternative methods of computing astonishingbecauseshaperesonancesofH+ arenotonly
si the energy-level structure of H+2 and H2 not relying on intrinsicallyinterestingbutalsobecauseth2eyrepresenta
y theBorn-Oppenheimerapproximationhavealsobeende- channelfortheformationofH+ inH+ +H(1s)collisions
h 2
veloped [18–22]. High-order perturbative calculations of by radiative or three-body recombination.
p
[ relativistic and radiative corrections have been reported Wereporttheobservationofthelowest-N+ shaperes-
1 boHth+f,oHrDH++2,[H23–,2a6n]danHdDHa2re[6,am27o]n.g the first molecules Hon+anacneds otfhHe+2(1,7t,h7e)(rve+so=na1n8c,eNo+f =ort4h)oreHso+naanncdeporfepsaenrat
v 2 2 2 2
3 tohavebeenformedintheuniverseandarethereforealso calculations of their positions and widths using Born-
0 ofcentralimportanceinastrophysics. Throughreactions Oppenheimerpotential-energyfunctionsoftheX+ state
6 with H , the most abundant molecule in the interstel- [7, 47–49], and adiabatic [7, 49], nonadiabatic [16], rela-
2
3 lar medium, H+ is converted into H+, so that H+ has tivistic and radiative corrections [12, 50, 51].
0 notbeendetect2edinastrophysicalspe3ctrasofar[282,29], The quasi-bound levels of the X+ state of H+ were
. 2
1 despite extensive searches by radioastronomy. studied by pulsed-field-ionization zero-kinetic-energy
0
481 and 4 rovibrational levels are believed to exist (PFI-ZEKE) photoelectron spectroscopy [52] using an
7
1 in the ground (X+ 2Σ+g) and first excited (A+ 2Σ+u) electric-field pulse sequence designed for high spectral
: electronic states of H+, respectively [12, 30–32], but resolution[53]. Thespectrawereobtainedbymonitoring
v 2
theelectronsproducedbyfieldionizationofveryhighRy-
i only a fraction of these have been observed experi-
X dberg states (principal quantum number n beyond 100)
mentally, using methods as diverse as microwave elec-
r tronic [33–35] and pure rotational [36] spectroscopy, located below the ionization thresholds of H2 as a func-
a tion of the wave number of a tunable laser. To access
radio-frequency spectroscopy of magnetic transitions be-
the bound and quasi-bound rotational levels of the high-
tween fine- and hyperfine-structure components [37, 38],
est vibrational states (v+ = 16−19) of the X+ state of
photoelectron spectroscopy [39–41], and Rydberg-state
H+ from the X 1Σ+ ground state of H , a three-photon
spectroscopy combined with Rydberg-series extrapola- 2 g 2
excitation sequence
tion [42–46]. The main reason for the incomplete ex-
perimental data set on the level structure of H+ is the
2 H+ ←V−I−S−2 H¯(11,2-3)←V−I−S−1 B(19,1-2)←V−U−V−X(0,0-1).
absence of allowed electric-dipole rotational and vibra- 2
+PFI
tional transitions. (1)
Of the 481 rovibrational levels of the X+ state of H+,
2
58 are known to be quasi-bound tunneling (shape) reso- wasusedviatheB 1Σ+ (19,1or2)andtheH¯ 1Σ+ (11,2
u g
2
and the selectivity of the PFI process. The instrumen-
145815 A+ tal line-shape functions adequate to describe the spec-
tra recorded with the successive pulses of the PFI se-
145810 (17,7)
quence (see inset of Fig. 2(a)) are Gaussian functions
-1m)145805 withafullwidthathalfmaximum(FWHM)of0.2cm 1
c −
0) (145800 (18,4) (pulses 2 to 5 in the pulse sequence), 0.25 cm−1 (pulse
e X(0,145795 (19,1) (0,1) (0,2) o6f),thanedsp0e.c3t5racmre−c1ord(peudlswesith7 paunldse8s).9 aTnhde1l0inweewreidttohos
v (0,0)
o145790 (18,3) large, and the signal recorded with the first pulse was
b
a too weak, to be included in the analysis. Mass-analyzed
er 145785 (16,9)
mb (18,2) threshold ionization (MATI) spectra [58] were recorded
u145780 (17,6) X+ with a pulse sequence consisting of a discrimination
n
e (18,1) pulse of −70 mV/cm followed by an extraction pulse of
v
Wa126500 (11,3) 800 mV/cm and monitoring H+ and H+2 ions.
124500 _ Figures 2(a) and 2(b) display the PFI-ZEKE photo-
H H electron spectra of para and ortho H in the vicinity of
122500 2
0 5 10 15 20 25 30 the H+ + H(1s) + e DI threshold recorded from the
−
R (a0) H¯ 1Σ+ (11,2 and 3) intermediate levels, respectively.
g
Ten spectra were recorded simultaneously by monitor-
FIG.1. Potential-energyfunctionsoftheHH¯ stateofH [56]
2 ing the electrons produced by the ten electric-field steps
(lower panel) and the X+ (solid, dashed) and A+ (dotted)
states of H+ [7, 48]. Selected vibrational wave functions and of the pulse sequence (see inset of Fig. 2(a)) but only
2
energylevelsofpara(red)andortho(blue)hydrogenaredis- five, corresponding to the steps labeled A-E, are shown
played. forclarity. Theupperhorizontalscaleindicatesthewave
number above the H (v =0,N =0) ground state, which
2
was determined from the known term value of the se-
or 3) intermediate levels. Selecting vibrational levels of lected H¯ 1Σ+ (11,N = 2 or 3) level [55] and the wave
g
the outer (H¯) well of the HH¯ state is ideal for access- number ν˜ (see Eq. (1)). The scale given below each
VIS2
ing long-range states of molecular hydrogen, as demon- spectrum in Figs. 2(a) and 2(b) gives the wave num-
strated by Reinhold et al. [54, 55], who also reported the ber relative to the positions of the X+(17,6) and the
absolute term values of many rovibrational levels of the X+(18,3) states, respectively. The position of the DI
H¯ state. Fig. 1 depicts the potential-energy functions threshold, 145796.8413(4) cm 1 [45, 59] is marked by a
−
of the HH¯ state [56] of H (lower panel) and the X+ grey dashed vertical line and coincides with the onset
2
(N+ = 0,4 and 7) and A+ states of H+ [7, 48] (upper of a continuum in the spectra. Because the spectra dis-
2
panel) and selected vibrational wave functions. The fig- play the yield of electrons produced by delayed PFI, the
ure indicates that the v = 11 vibrational level of the H¯ electron signal measured in the continuum must stem
can be used to access the ionization continua associated from the field ionization of high-n Rydberg states of H,
withthehighestvibrationallevelsoftheX+ andthefew a conclusion that was confirmed by the MATI spectra,
bound levels of the A+ state. displayed in the upper part of Fig. 2(b).
The vacuum-ultraviolet (VUV) radiation around ThespectraofparaH (Fig.2(a))consisteachofthree
2
105680cm 1 neededinthefirststepoftheexcitationse- sharplineslocatedbelowtheDIthreshold, whichcanbe
−
quence(1)wasgeneratedbyfour-wavemixinginapulsed unambiguously attributed to the (17,6) and (18,2) levels
beam of Kr gas using two Nd:YAG-pumped pulsed dye of the X+ 2Σ+ ground state and the (0,1) level of the
g
lasers (pulse duration 5 ns), as described in Ref. [57]. A A+ 2Σ+ first excited state. The spectra also reveal a
u
third pulsed dye laser was used to access the H¯ (v =11) broader line above the DI threshold. Modeling the line
levels from the selected levels of the B state. A fourth shapebytakingintoaccounttheexperimentalline-shape
tunablepulseddyelaser,delayedbyapproximately10ns function and after subtraction of the contribution of the
with respect to the other two laser pulses, was used to DI continuum indicates a Lorentzian line-shape function
access the region near the dissociative ionization (DI) with a FWHM of 0.21(7) cm 1, which suggests that this
−
threshold of H . level is a quasi-bound level of H+. Based on the calcula-
2 2
All three laser beams intersected a pulsed skimmed tions presented below, we assign this line to a transition
supersonic beam of neat H at right angles on the axis to the quasi-bound (18,4) level of H+.
2 2
of a PFI-ZEKE photoelectron spectrometer [57]. The The spectra of ortho H (Fig. 2(b)) also reveal transi-
2
wave number of the fourth dye laser was calibrated at tionstoboundandquasi-boundlevelsofH+. Thebound
2
an accuracy of 600 MHz (3σ) using a wave meter. The statesofH+ observedinthesespectraareassigned,inor-
2
resolution of the photoelectron spectra was determined der of increasing energy, to the X+ (18,1), (16,9), (18,3)
bythebandwidthofabout1GHzofthefourthdyelaser levels,theA+ (0,0),(0,2)levels,andtheX+ (19,1)level,
3
WavenumberaboveX(v=0,N=0)(cm−1) WavenumberaboveX(v=0,N=0)(cm−1)
145770 145780 145790 145800 145770 145780 145790 145800 145810
t
45..50 m)0.00-0-.00-.050-.070.80A9B 45..50 x1 x30 HH++2
units)4.0 F(V/c -0-.01-.011-.042.02C7DE 4.0 -20 -15 -10 -5 0 5 10 15 20 25
(arb.3.5 E -0-.16.822 units)3.5 E
gnal3.0 -5 0 5 10 15 20 25 arb.3.0 -20 -15 -10 -5 0 5 10 15 20 25
ctronsi2.5 (17,6) (18,2) (0,1) (18,4) signal(2.5 (18,1) (16,9) (18,(30),0)(0(1,29),1) (17,7)
KEphotoele12..50 -5 CD 0 5 10 15 20 25 FI-fragment12..50 -20CD -15 -10 -5 0 5 10 15 20 25
E P
Z1.0 -5 0 5 10 15 20 25 1.0 -20 -15 -10 -5 0 5 10 15 20 25
FI- B B
P
0.5 0.5
-5 0 5 10 15 20 25 -20 -15 -10 -5 0 5 10 15 20 25
A A
0.0 0.0
-5 0 5 10 15 20 25 -20 -15 -10 -5 0 5 10 15 20 25
[Wavenumber-ν˜(v+=17,N+=6)](cm−1) [Wavenumber-ν˜(v+=18,N+=3)](cm−1)
(a) (b)
FIG.2. PFI-ZEKEphotoelectronspectraofpara(a)andortho(b)H inthevicinityoftheDIthreshold,whichisindicatedby
2
a grey vertical dashed line. The spectra were recorded with the electric field steps labeled A-E in the pulse sequence depicted
in the inset in (a). MATI spectra for ortho H are displayed in the upper trace in (b).
2
thepositionofwhichislocatedjustbelowtheDIthresh- the predissociation widths of these levels are of the same
old. ThebroaderlineobservedabovetheDIthresholdis magnitude as the resolution of our experiment, and (2)
attributedtoasecondquasi-boundrotationallevelofthe the DI-continuum cross section is not known and is thus
X+ state, the (17,7) level, for which we derive by decon- difficulttocleanlyremovebysubtraction. Thisdifficulty
volution a Lorentzian line-width function with a FWHM alsohinderedthequantitativeanalysisofthe(19,1)level,
of 0.56(8) cm 1. This conclusion is confirmed by the which is therefore omitted in Table I.
−
fact that this line appears in the H+ mass channel of the
The observation of transitions to states of high rota-
MATI spectrum (see Fig. 2(b)). tional quantum number N+, up to N+ =9 in ortho H+
2
The positions of the DI threshold and of the H+ + in Fig. 2(b), is attributed to the fact that, at long range,
2 theH¯ statehasH+H ion-paircharacter. Consequently,
e ionization thresholds gradually shift towards lower −
−
a single-center expansion of the orbital out of which the
wave numbers at successive steps of the field-ionization
electron is ejected consists of several (cid:96) components ((cid:96) is
sequence, with shifts of −0.68(5), −0.81(5), −1.07(5),
−1.27(5), and -1.59(5) cm 1 for the pulses A-E. These the orbital angular momentum quantum number). Ap-
−
plyingphotoionizationselectionrules[60,61]leadstothe
shifts are equal to those we determine in simulations
conclusion that |∆N| = |N+−N| must be equal to, or
of the PFI dynamics using the method described in
less than, (cid:96) + 2, where (cid:96) is the highest compo-
Ref.[53]. Whengivenrelativetothe(17,6)andthe(18,3) max max
nent in the single-center expansion of the H¯ orbital, and
thresholds in Figs. 2(a) and 2(b), respectively, the posi-
that ∆N must be even (odd) for the X+(A+)←H¯ pho-
tions of the lines in the spectra recorded with different
toionizingtransition. ThespectrapresentedinFigs.2(a)
pulse steps are identical within the experimental uncer-
and 2(b) indicate that (cid:96) is at least 4.
tainties because the PFI shifts are exactly compensated. max
The positions of the levels of para and ortho H+ deter- In Fig. 2, the relative intensity of the photoelectron
2
mined experimentally are listed relative to the positions signal in the continuum compared to that below the DI
of the (17,6) and (18,3) levels in Table I, where they are threshold increases at each successive step of the field-
compared with the theoretical values of Moss [12] and ionization sequence. This trend can be explained in part
the results of our own calculations. Whereas the relative bythefactthattheresolutionofthephotoelectronspec-
positions of all levels of H+ could be determined with tra decreases at each step of the pulse sequence, with
2
uncertainties of only 0.06 cm 1, the uncertainties in the the consequence that sharp structures are less efficiently
−
widths of the quasi-bound levels are larger because (1) excited than broad ones.
4
using a fifth-degree Lagrange polynomial and all func-
TABLE I. Positions of the observed (o) bound and quasi- tions were smoothly connected to the H++H(1s) disso-
bound states of H+ compared with the calculated (c) values.
Thelevelsofpara2andorthoH+ aregivenwithrespecttothe ciation limit. The energy Eres and the FWHM Γ of the
2
X+(17,6) and (18,3) levels, respectively. The experimental resonances were determined by calculating the energy-
uncertainties represent one standard deviation. dependent phase shift δN+(E) for each N+ [65]. Be-
cause lim Uad(R)=const., the asymptotic solution
R
Level ν˜o(cm−1)a o−c(cm−1)b o−c(cm−1)a of Eq. (2) →is∞a linear combination of the regular and ir-
(17,6) 0 0 0 regularsphericalBesselfunctionsj (kR)andn (kR)
N+ N+
(18,2) 4.349(27) 0.0267 0.0406 with k2 = 2µ(E−Uad). The phase shift for a given en-
(0,1) 16.08(4) 0.0161 − ergy δ (E) was obtained by using the values of the
N+
(18,4) 20.41(4)c − 0.0191d wave function at the two outermost grid points R and
a
(18,1) -14.56(4) 0.0193 0.0107 Rb =Rmax using
(16,9) -6.39(6) -0.0035 -0.0255
(18,3) 0 0 0
Kj (R )−j (R ) R χ (R )
N+ a N+ b a N+ b
(0,0) 2.61(3) 0.0168 − tanδN+ = Kn (R )−n (R ); K = R χ (R ).
(0,2) 5.265(16) 0.0404 − N+ a N+ b b N+ a
(3)
(17,7) 17.11(6)e − -0.0105 f
For an isolated resonance in a single channel the energy
aThis work. bRef. [12]. cΓ =0.21(7) cm−1. dependence of the phase shift is given by
o
dΓ =0.20 cm−1. eΓ =0.56(8) cm−1. fΓ =0.16 cm−1.
c o c
tan(cid:2)δ (E)−δ0 (cid:3)= Γ , (4)
Rovibrational energies Ei and the nuclear wave func- N+ N+ 2(Eres−E)
tions χ (R) were calculated in atomic units by solving
i
(cid:20) 1 d2 N+(N++1) (cid:21) where δN0+ is assumed to be constant near Eres.
− +Uad+ −E χ (R)=0,
2µ dR2 2µ R2 i i The experimental positions of bound and quasi-bound
vib rot
(2) levels of H+ agree with the calculated positions within
2
where Uad = UCN+H +H is the adiabatic potential the experimental uncertainty of 0.06 cm 1. The po-
1 2 −
curve with the clamped-nuclei energy UCN = Uel+1/R sitions of the bound levels we calculate with our ef-
and the electronic energy Uel is obtained by solving the fective potential agree with the results of Moss within
electronic Schr¨odinger equation at fixed R. UCN and 0.025cm 1andweattributethedifferencestotheincom-
−
the adiabatic corrections H1 = −21µ(cid:82) ψi∗∆Rψidr and pletedescriptionofthenonadiabaticeffectsinthepresent
H2 = −81µ(cid:82) ψi∗∆rψidr were taken from [7, 49]. Be- work. The width we observe for the (18,4) quasi-bound
level agrees with the calculated value, but the measured
cause the X+ state is well separated from other ger-
width of the (17,7) resonance is more than three times
ade states, the leading term of the nonadiabatic correc-
larger than the width we calculate. Given the excellent
tions can be evaluated conveniently by introducing R-
agreement of the positions, we do not have a good ex-
dependent reduced masses for vibration and rotation,
planation for this discrepancy. It may simply be a con-
which allows one to retain the idea of a single elec-
sequence of the approximate nature of our calculations.
tronic potential function [15, 16]. Vibrational and ro-
tational masses µ−vi1b = µ−1(1+A(R)/mp) and µ−ro1t = AorlttheornHati,vewlyhitchheisiomnsorgeentehraantetdhrienetthiemDesIsctornotnigneurutmhaonf
µ 1(1+B (R)/m ) were determined using A(R) and 2
− pol p
in para H , may broaden the PFI-ZEKE signal. The
B (R) as given in [16]. The proton-to-electron mass 2
pol
discrepancy may further indicate nonadiabatic interac-
ratio was taken to be m /m = 1836.15267389(17) and
p e tions in the three-body system H(1s)-e -H+ which, in
E /hc=2194746.313702(13) cm 1 [59]. The relativistic −
h −
thisenergyregion,maydecayeitherbyionizationordis-
and radiative corrections as reported by Moss [12] were
sociation with chaotic branching ratios. The fact that
added to our nonadiabatic energies.
thefield-ionizationshiftsbehavenormallyspeaksagainst
We implemented the renormalized Numerov method
the latter two explanations. Further theoretical work is
as described in [62] to solve Eq. (2) numerically on a
needed to clarify this discrepancy.
grid (0.2 a ,R = 200 a ) with an integration step
0 max 0
of 0.01 a . UCN was interpolated with a fifth-degree We thank Dr. Ch. Jungen (Orsay) for useful discus-
0
polynomial that fits Uel and dUel/dR simultaneously sions. The content of this letter is related to material
at three points [63]. dUel/dR was calculated from Uel presented in May 2015 during the Kolos Lecture at the
and the adiabatic correction H using the virial theorem Department of Chemistry, University of Warsaw. This
2
which holds exactly within the Born-Oppenheimer ap- work is supported financially by the Swiss National Sci-
proximation [64]. The other functions were interpolated ence Foundation under project SNF 200020-159848.
5
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