High-resolution millimeter wave spectroscopy and

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Dec 15, 2004 - Table I provides a summary of all quantum numbers used in the .... Projection of Rydberg orbital angular momentum on internuclear ... ground state to channels associated with the v. 0 levels of ...... 5d. 3. 0 b. 2. 13c. 30 d. 3. 0. 0. 0. 0. 0 b. 2 c. 10 d. 2 . B4. This matrix was used to derive Eqs. 17–21 in Sec. V.
JOURNAL OF CHEMICAL PHYSICS

VOLUME 121, NUMBER 23

15 DECEMBER 2004

High-resolution millimeter wave spectroscopy and multichannel quantum defect theory of the hyperfine structure in high Rydberg states of molecular hydrogen H2 A. Osterwalder,a) A. Wu¨est, and F. Merktb) Physical Chemistry, ETH Zurich, CH-8093 Zurich, Switzerland

Ch. Jungenc) Laboratoire Aime´ Cotton du CNRS, Universite´ de Paris Sud, F-91405 Orsay, France

共Received 14 May 2004; accepted 23 July 2004; publisher error corrected 10 December 2004兲 Experimental and theoretical methodologies have been developed to determine the hyperfine structure of molecular ions from detailed studies of the Rydberg spectrum and have been tested on molecular hydrogen. The hyperfine structure in ᐉ⫽0 – 3 Rydberg states of H2 located below the ⫹ ⫹ ⫹ X 2⌺ ⫹ g ( v ⫽0,N ⫽1) ground state of ortho H2 has been measured in the range of principal quantum number n⫽50– 65 at sub-MHz resolution by millimeter wave spectroscopy following laser excitation to np and nd Rydberg states using a variety of single-photon and multiphoton excitation sequences. The np1 1 , nd1 1 , and the n f 1 0 – 3 Rydberg states were found to be metastable and to have lifetimes of more than 5 ␮s beyond n⫽50. Members of other series, such as the nd1 2 , nd1 3 , and the np1 0 series, were found to have lifetimes of more than 1 ␮s. Local perturbations induced by low-n Rydberg states belonging to series converging on rovibrationally excited levels of H⫹ 2 reduce the lifetimes in narrow ranges of n values. The hyperfine structure is strongly dependent on the value of the orbital angular momentum ᐉ. In the penetrating s and p states at n⬇50 the exchange interaction dominates over the hyperfine interaction and the levels can be labeled by the total electron spin angular momentum quantum number S (S⫽0 or 1兲. In the less penetrating d and f Rydberg states, the hyperfine interaction between the core nuclear and electron spins is larger than the exchange interaction and the Rydberg states are of mixed singlet and triplet character. A procedure based on the Stark effect and on the systematic analysis of selection rules and combination differences was developed to determine the orbital and the total angular momentum quantum numbers ᐉ and F and to construct an energy map of p and f Rydberg levels between n ⫽54 and 64 with relative positions of an accuracy of better than 1 MHz. Multichannel quantum defect theory 共MQDT兲 was extended to treat the hyperfine structure in molecular Rydberg states and was used to analyze the observed hyperfine structure of the p and f Rydberg states of H2 . The frame transformation between the Born-Oppenheimer channels described by the angular momentum coupling scheme (a␤ J ) and the asymptotic channels described by the (e关 b␤ S ⫹ 兴 ) coupling scheme was derived and enables an elegant treatment of all intermediate coupling cases. Purely ab initio quantum defect theory reproduced the experimentally determined positions to within 40 MHz for the p levels and 13 MHz for the f levels. By slight adjustments of the quantum defect functions and their energy dependences and by consideration of the p- f interaction, of the singlet-triplet splittings of the f levels, and of the departure of the ionic levels from pure coupling case (b␤ S ⫹ ), the agreement between theory and experiment could be improved to 600 kHz. By comparing the results of MQDT calculations of the hyperfine structure of f Rydberg levels with those of coupled equations calculations, the frame transformation approximation of MQDT was shown to be accurate ⫹ ⫹ to within 300 kHz. The extrapolated ionic hyperfine structure of the X 2 ⌺ ⫹ g ( v ⫽0,N ⫽1) ionic level corresponds to the ab initio prediciton of Babb and Dalgarno 关Phys. Rev. A 46, R5317 共1992兲兴 within the experimental error. © 2004 American Institute of Physics. 关DOI: 10.1063/1.1792596兴

I. INTRODUCTION

This work summarizes the results of a joint experimental and theoretical investigation of the hyperfine structure in p and f Rydberg states of molecular hydrogen with principal a兲

Present address: Department of Chemistry, University of California, Berkeley, California 94720. b兲 Author to whom correspondence should be addressed. c兲 Author to whom correspondence should be addressed. 0021-9606/2004/121(23)/11810/29/$22.00

quantum number in the range n⫽54– 64 belonging to series ⫹ converging on the hyperfine components of the X 2 ⌺ ⫹ g (v ⫹ ⫹ ⫽0,N ⫽1) level of the ortho H2 ion. The experimental results were obtained with a spectral resolution of 300 kHz by millimeter wave spectroscopy and the spectra were analyzed by multichannel quantum defect theory 共MQDT兲. The current knowledge on the fine and hyperfine structure in electronically highly excited states of molecules is limited to a few triplet s, p, and d Rydberg states with n

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J. Chem. Phys., Vol. 121, No. 23, 15 December 2004

⫽2–4 and several ᐉ⫽3 – 8 nonpenetrating Rydberg states with n⫽4,5,10 and 27 of molecular hydrogen. Miller and Freund analyzed the fine and hyperfine structure of the 3 ⌸ u (n⫽2 – 4) Rydberg states in para and ortho H2 . 1 Lichten and co-workers studied the hyperfine structure of the 3p d 3 ⌸ u and 4p k 3 ⌸ u Rydberg states2,3 and Ottinger and co3 ⫹ 3 3 workers that of the g 3 ⌺ ⫹ g , h ⌺ g , i ⌸ g , and j ⌬ g states 4–6 Lundeen and co-workers have measured (ᐉ⫽0 – 2,n⫽3). the hyperfine structure in nonpenetrating (ᐉ⫽4 – 8) Rydberg states at n⫽10 and 27, and used a long-range corepolarization model to extract the hyperfine structure of the 7–13 ortho H⫹ Uy et al.14 were the first to extend and 2 ion. apply MQDT to treat the hyperfine structure in molecular Rydberg states and reported on the analysis of the 5g-4 f transition. The results of the MQDT calculations, which were in good agreement with the experiment, suggested that a considerable wealth of spectral structures might become observable by high-resolution spectroscopy: each rotational line of the 5g-4 f band, observed as a single line at an experimental resolution of 0.1 cm⫺1 , was predicted to consist of no less than 83 transitions. Recently, a systematic measurement of the hyperfine structure in ᐉ⫽0 – 3 Rydberg states of H2 was carried out by millimeter wave spectroscopy in the range of principal quantum number n⫽50– 65. 15,16 The spectral resolution of up to 300 kHz enabled the observation of the complete hyperfine structure. This study illustrated the rapid uncoupling of the Rydberg electron from the H⫹ 2 core motion at increasing values of ᐉ: in a p Rydberg state at n⬇55, the exchange interaction between Rydberg and core electrons is still sufficiently large so that the Rydberg states can be characterized by the total electron spin, S⫽0 or S⫽1. The energy level structure within the S⫽0 and the S⫽1 manifolds of states can each be qualitatively described by ᐉ-uncoupling theory. In the nonpenetrating f Rydberg states at n⫽55, and to a lesser extent also in the d Rydberg states, the hyperfine interaction between the core electron spin and the nuclear spin is dominant over the exchange interaction and the Rydberg states are of mixed singlet and triplet character. The mixed character of the d Rydberg states is of particular interest for experimental investigations of both the singlet and the triplet manifolds of the p Rydberg states. In this contribution, a quantitative comparison between experimental results and the results of MQDT calculations is presented for the n p and n f Rydberg states. A later report will be dedicated to the ns and nd Rydberg states which present additional complications caused by the more penetrating nature of the Rydberg electron and perturbations caused by doubly excited states.17 The main motivations for this work were the following: 共a兲 To reach a quantitative understanding of the hyperfine structure of highly excited electronic states of H2 by a systematic comparison of experimental results and theoretical calculations. 共b兲 To assess the limits of MQDT calculations and identify where improvements may be necessary; in particular, to assess the predictive power of purely ab initio MQDT. The experimental data, with the completely resolved hyperfine

Rydberg states of molecular hydrogen

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structure, provide a very stringent test of the theory: having to account for spectral positions of electronically excited states with a precision of ⬇300 kHz indeed is a challenge for any theory of electronic structure. 共c兲 To try and identify possible sources of experimental artefacts and confirm the experimental assignment of the relevant quantum numbers by comparison with theoretical results. 共d兲 To improve, in a least-squares fitting procedure, some of the MQDT parameters and thus obtain a more accurate description of the ionization channels of H2 . 共e兲 To provide, by extrapolation, an energy map reaching all the way to the ionization threshold to assist future highprecision determinations of the ionization potential of H2 . The most promising strategy appears to be a series of three distinct measurements of the three successive energy inter18 1 ⫹ vals between the X 1 ⌺ ⫹ between g and the E,F ⌺ g states, ⫹ the E,F 1 ⌺ g states and p or f Rydberg states of principal quantum number n⬇50, and finally between these Rydberg states and the ionization limit. We show here that the combination of MQDT and millimeter wave spectroscopy provides, for the last interval, an absolute accuracy of better than 1 MHz. The combination also yields, by extrapolation, the hyperfine structure of the H⫹ 2 ion, although the precision reached here does not appear to be as high as that reported previously by Lundeen and his co-workers.13 Table I provides a summary of all quantum numbers used in the present work to characterize the various levels of H2 and H⫹ 2 . A ‘‘⫹’’ superscript is used to distinguish the quantum numbers of the cation from those of the neutral molecule. In the present situation, the total nuclear spin quantum number is I⫽0 for para H2 and I⫽1 for ortho H2 , 19 the electron spin quantum number is S ⫹ ⫽1/2 in H⫹ 2 and S⫽0 or 1 in H2 , the total spin quantum number G ⫹ takes values of 1/2 and 3/2, ⌳ ⫹ ⫽0 and ⍀ ⫹ ⫽1/2. When (⫹) (⫹) necessary, the usual term symbols (2S ⫹1) ⌳ ⍀ (⫹) (g/u) are also employed, where the symbols g and u refer to the symmetry of the electronic wavefunction with respect to inversion at the molecular center of mass and the optional ⫹ superscript is used to label the quantum states of the ion. F (⫹) and M F (⫹) represent the only good quantum numbers next to the parity index p (⫹) which takes values of 0 and 1 for states of positive and negative parity, respectively. We complete this introduction by summarizing the different angular momentum coupling schemes used to describe the hyperfine structure of molecular Rydberg states. As long as the nuclear spin is disregarded, the usual Hund’s coupling cases can be used whereby it is often necessary to distinguish between the coupling case of the ionic core to which the Rydberg electron is attached and that of the Rydberg state as a whole 共see Refs. 20 and 21 and references therein兲. For ⫹ instance, the X 2 ⌺ ⫹ g ground electronic state of H2 is well described by Hund’s case 共b兲, and the Rydberg states of H2 in the close-coupling range of the electron-core ‘‘collision’’ complex or at low n values by either Hund’s cases 共a兲 or 共b兲. In this region, the Rydberg electron is still coupled electrostatically to the internuclear axis and can be labeled as nᐉ␭. The Rydberg states of H2 are then designated by nᐉ (2S⫹1) ⌳ ⍀(g/u) where S⫽0 or 1, ⌳⫽␭, and ⍀⫽⌳,⌳

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TABLE I. Quantum numbers and symmetry labels used in the present work to describe the hyperfine structure in Rydberg states of diatomic molecules.

Physical quantity Total angular momentum F Total angular momentum without spins N Total angular momentum without nuclear spins J Total angular momentum without Rydberg electron spin Fs Total electron spin S Total nuclear spin I Total Rydberg electron angular momentum j Rydberg electron orbital angular momentum 艎 Rydberg electron spin s Total core angular momentum F⫹ Total core angular momentum without spinsa N⫹ Total core angular momentum without nuclear spins J⫹ Total core-electron spin S⫹ Total core spin G⫹ Projection of angular momentum i on space-fixed z axis Projection of total orbital angular momentum on internuclear axis Projection of Rydberg orbital angular momentum on internuclear axis Projection of core orbital angular momentum on internuclear axis Projection of total electronic angular momentum on internuclear axis Projection of core electronic angular momentum on internuclear axis Parity index Ion core parity index Electronic symmetry indexb Electronic symmetry index of ion coreb Inversion symmetry of the electronic wavefunction a

Quantum number F N J Fs S I j ᐉ s F⫹ N⫹ J⫹ S⫹ G⫹ Mi ⌳ ␭ ⌳⫹ ⍀ ⍀⫹ p p⫹ q q⫹ g/u

In a ⌺ state, this is equivalent to the rotational angular momentum. q and q ⫹ take the value 0 except in ⌺ ⫺ states where they take the value 1.

b

⫾1. In the asymptotic range, and at high n values, Hund’s case 共d兲 limit is approached in which ᐉ is coupled to the core rotation N⫹ to form N⫽N⫹ ⫹ᐉ, and the Rydberg states are labeled nᐉN ⫹ N . Many additional angular momentum coupling cases arise when the nuclear spin is considered. The convention used to describe these coupling cases is to indicate by an additional subscript to the usual labels 共a兲, 共b兲,..., how the nuclear spin is coupled.22,23 For instance, (b␤ J ) and (b␤ S ) indicate electronic states approaching Hund’s case 共b兲 in which the total nuclear spin I is not coupled to the internuclear axis 关‘‘␤’’ in analogy to ‘‘共b兲’’兴 but to J and S, respectively. Similarly, (a␤ J ) designates an electronic state approaching Hund’s case 共a兲 in which the nuclear spin is coupled to J. When labeling Rydberg states in this scheme, it is appropriate to treat the close-coupling range 共and the low-n states兲 and the asymptotic range 共and the high-n states兲 separately. In the former case, the interactions involving the nuclear spin are always much weaker than the electrostatic, spin-orbit or exchange interactions, and the explicit consideration of the nuclear spin does not justify a change of nomenclature compared to the situation without nuclear spin other than the specification of the total angular momentum quantum number F. 24 The corresponding angular coupling cases are thus

Osterwalder et al.

always ‘‘␤ J.’’ In the latter case, the completely decoupled 共asymptotic兲 situation requires the use of several versions of Hund’s case 共e兲 in which the total Rydberg electron angular momentum j⫽艎⫹s hardly interacts with the total core angular momentum F⫹ to form F. The diversity of Hund’s cases 共e兲 arises from the different ways the core angular momenta are coupled and are conveniently labeled 共e 关core coupling case兴兲. For example, (e关 a␤ J ⫹ 兴 ) is the suitable case to describe a Rydberg state in which the Rydberg electron is weakly coupled to an ion core in Hund’s case (a␤ J ⫹ ). The appropriate Hund’s case 共e兲 for the high Rydberg states of ortho H2 is (e关 b␤ S ⫹ 兴 ). This case implies an angular momentum coupling hierarchy characterized by a dominant interaction between the core spins S⫹ and I to form G⫹ , a weaker interaction between G⫹ and N⫹ to form F⫹ , and, finally, the weakest interaction between F⫹ and j to form F. This asymptotic limiting case is important to define the proper angular momentum frame transformation used in the MQDT treatment of the hyperfine structure but has not been encountered experimentally in the present study of Rydberg states in the range n⫽50– 65. Instead, two intermediate situations have been observed: In the penetrating s and p series, the exchange interaction between core and Rydberg electron is sufficiently large so that the core electron spin couples to the Rydberg electron spin and not to the nuclear spin. As a result, S remains an approximately good quantum number and Hund’s case (d␤ S ) limit is reached separately for the singlet and triplet states. The Rydberg states are then conveniently labeled nᐉN ⫹ N (S)F. In the nonpenetrating f Rydberg series, and to a lesser extent the d series, G⫹ is almost a good quantum number as in the limiting case (e关 b␤ S ⫹ 兴 ). However, the Rydberg electron angular momentum 艎 remains coupled to the core rotation N⫹ to form N which then couples to G⫹ to form Fs which finally couples to s to form F. The appropriate labeling scheme is thus nᐉN ⫹ N (G ⫹ )F. S is not a good quantum number here and the f 共and also the d) states have mixed singlet and triplet character.15 In both observed intermediate cases, it is important to realize that the only good angular momentum quantum number is F and that all others merely represent useful labels. The paper is structured as follows: Section II presents the experimental procedure in detail and quantifies the uncertainties and systematic errors in the determination of the transition frequencies and of the level positions. Section III gives an overview of the general energy level structure of p ⫹ and f Rydberg states in the vicinity of the X 2 ⌺ ⫹ g (v ⫹ ⫽0,N ⫽1) ionization threshold. Section IV provides a survey of the experimental results which comprise measurements of the dynamical behavior of s, p, d, and f Rydberg states by laser spectroscopy and the determination of the hyperfine structure in these Rydberg states by millimeter wave spectroscopy. The section also describes how the assignment of all relevant quantum numbers could be reached. The determination of these quantum numbers independently from the calculations was an important element in our strategy because it enabled a more stringent test of the theory. Section V is devoted to a discussion of the hyperfine structure of the

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J. Chem. Phys., Vol. 121, No. 23, 15 December 2004

Rydberg states of molecular hydrogen

⫹ ⫹ ⫹ X 2⌺ ⫹ g ( v ⫽0,N ⫽1) state of the H2 ion. Section VI summarizes the main features of the MQDT approach to study the hyperfine structure of Rydberg states with emphasis on the ungerade Rydberg states of H2 . Section VII finally compares the experimental results on the hyperfine structure of p and f Rydberg states of H2 with MQDT calculations and gives an overview of the information that can be extracted from such a comparison on the dynamical and structural properties of the Rydberg states of H2 and on the hyperfine structure of the H⫹ 2 ion.

II. EXPERIMENT

The experiments were carried out using tunable sources of VUV, UV and millimeter wave radiation to excite H2 molecules to high Rydberg states using several multiphoton excitation sequences. The H2 sample gas was introduced in the spectrometer in a skimmed supersonic expansion. The excitation was detected by monitoring the ionization signal resulting from photoionization or pulsed electric field ionization of long-lived Rydberg states. A. The excitation sequences

Three multiphoton excitation sequences, described by Eqs. 共1兲–共3兲, were used to study the low-ᐉ (ᐉ⫽0 – 3), highn Rydberg states of ortho H2 in the vicinity of the ⫹ ⫹ X 2⌺ ⫹ g ( v ⫽0,N ⫽1) ionization threshold: X 共 v ⵮ ⫽0,N ⵮ ⫽1,F ⵮ ⫽0 – 2 兲 →B 共 v ⬙ ⫽0,N ⬙ ⫽0,F ⬙ ⫽1 兲 →n ⬘ d 共 N ⬘ ⫽1,F ⬘ ⫽0 – 2 兲 →np/n f 共 N⫽0 – 2,F⫽0 – 3 兲 , 共1兲 X 共 v ⵮ ⫽0,N ⵮ ⫽1,F ⵮ ⫽0 – 2 兲 →B 共 v ⬙ ⫽3,N ⬙ ⫽0,F ⬙ ⫽1 兲 →n ⬘ d 共 N ⬘ ⫽1,F ⬘ ⫽0 – 2 兲 →np/n f 共 N⫽0 – 2,F⫽0 – 3 兲 , 共2兲 X 共 v ⵮ ⫽0,N ⵮ ⫽1,F ⵮ ⫽0 – 2 兲 →n ⬙ p 共 N ⬙ ⫽0 – 2,F ⬙ ⫽0 – 3 兲 →n ⬘ s/n ⬘ d 共 N ⬘ ⫽0 – 3,F ⬘ ⫽0 – 4 兲 .

共3兲

In the excitation sequences 共1兲 and 共2兲, ortho H2 was cleanly selected by fixing the frequency of the VUV laser at the 1 ⫹ position corresponding to a P共1兲 line of the X 1 ⌺ ⫹ g -B ⌺ u transition. The selection of ortho H2 with excitation sequence 共3兲 was possible because the N ⫹ ⫽1 level of the ion is separated by ⬇60 cm⫺1 from the N ⫹ ⫽0 level and by 120 cm⫺1 from the N ⫹ ⫽2 level, i.e., the separations are half as large as in the neutral ground state. Consequently, transitions from the N ⵮ ⫽1 level of the ground state to members of the n ⬙ p(N ⬙ ⫽0 – 2) series converging on N ⫹ ⫽1 are spectrally well separated, at the high n values of interest in this study, from transitions from the N ⵮ ⫽0 to members of the n ⬙ p(N ⬙ ⫽1) series converging on N ⫹ ⫽0 and 2. The Franck-Condon factors for excitation from the ground state to channels associated with the v ⫹ ⫽0 levels of

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the ion via vibrational levels of the B state are such that excitation sequence 共2兲 is more than one order of magnitude more efficient than excitation sequences 共1兲 and 共3兲. The advantage of excitation sequence 共1兲, which was essential in the early phase of the experiment, is that the VUV radiation required for pumping the B-X transition lies well above the LiF cutoff wavelength 共105 nm兲 and can be generated by third harmonic generation of a single UV laser beam of wavelength ␭⬇330 nm. The VUV radiation around 94 000 cm⫺1 required to pump the B-X(3-0) transition in excitation sequence 共2兲 is more conveniently generated by resonance-enhanced sumfrequency mixing in a pulsed beam of Xe, but requires the use of a second dye laser. An increase of signal levels by more than two orders of magnitude could be reached with this latter excitation scheme, which enabled us to dilute the hydrogen gas in a H2 :Kr 1:50 gas mixture. The much slower velocity 共by a factor of 6 –7兲 of the supersonic expansion of this mixture compared to an expansion of neat H2 led to the following three advantages: 共i兲 a reduction of Doppler shifts, 共ii兲 a better confinement of the experimental volume probed in the millimeter wave experiments and consequently to a better control of the stray electric fields, and 共iii兲 an increase of the transit time of the molecules through the zone of interaction with the millimeter waves resulting in the possibility of increasing the measurement time. These advantages turned out to be essential to reach a spectral resolution of better than 1 MHz and to minimize the experimental uncertainties in the determination of the millimeter wave transition frequencies. Excitation sequence 共3兲 was introduced to obtain highresolution information on the ns1 1 Rydberg states by millimeter wave spectroscopy after it turned out that the intensity of this series was too weak in the laser spectra measured with sequences 共1兲 and 共2兲 for millimeter wave spectra to be recorded. Depending on the excitation sequence, two 关for sequences 共1兲 and 共3兲兴 or three 关for sequence 共2兲兴 dye lasers 共Lambda Physik, Scanmate 2E兲 pumped by the 532 nm or 355 nm outputs of an injection-seeded Nd:YAG 共YAG— yttrium aluminum garnet兲 laser 共Continuum, Powerlite 9010, repetition rate 10 Hz兲 were used. Tunable radiation to pump the B-X共0-0兲 transition was generated by third harmonic generation in a 15 cm long krypton tripling cell. The input UV beam (␭⬇330 nm, pulse energy of 1 mJ, pulse length 5 ns兲 was generated by doubling the frequency of a dye laser pumped by the 532 nm output of the Nd:YAG laser in a ␤ barium borate crystal and was focused by a 15 cm focal length lens in the tripling cell filled with the nonlinear gas. A LiF lens placed at the exit of the cell was used to recollimate the diverging VUV beam and, when traversed off center by the UV and VUV beams, to separate both beams. A metallic plate with a 1 mm diameter hole placed at the entrance of the photoexcitation chamber was used to block the undesired UV fundamental beam and transmit the VUV radiation. Phase matching at the wave numbers of the three lines 关 P(1), R(0), and R(1)] of the B-X 共0-0兲 band was achieved by mixing Kr with Ar. Satisfactory VUV intensities could be attained using neat krypton at a pressure of about 30 mbar,

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but the Kr:Ar mixing ratio and the pressure of the mixture were optimized, when necessary, to enhance the VUV intensity at the desired wavelengths by more than an order of magnitude. The VUV radiation around 94 000 cm⫺1 was generated by two-photon resonance-enhanced sum-frequency mixing (˜␯ vuv⫽2˜␯ 1 ⫹˜␯ 2 ) in a pulsed beam of xenon using the 5 p 5 ( 2 P 3/2)6p 关 1/2兴 0 ←5 p 6 1 S 0 two-photon resonance at 2˜␯ 1 ⫽80 118.97 cm⫺1 . The beam at ˜␯ 1 ⬇40 059 cm⫺1 was generated by doubling the frequency of a dye laser pumped by the third harmonic 共355 nm兲 of the Nd:YAG laser. The second beam of wave number ˜␯ 2 corresponded to the fundamental of a dye laser pumped by the second harmonic 共532 nm兲 of the same Nd:YAG laser. The sum-frequency beam was separated from other beams using a vacuum monochromator equipped with a gold-coated toroidal dispersion grating which also served the purposes of recollimating the diverging VUV beam and redirecting it toward the photoexcitation region where it crossed the skimmed supersonic expansion of the H2 probe gas at right angles. The tunable radiation around 124 500 cm⫺1 used in excitation sequence 共3兲 was generated by resonance-enhanced sum-frequency mixing in a pulsed beam of Kr using the 4 p 5 ( 2 P 3/2)5p 关 1/2兴 0 ←4 p 6 1 S 0 resonance at 2˜␯ 1 ⫽94 092.86 cm⫺1 . The beam of wave number ˜␯ 2 around 30 000 cm⫺1 was produced by frequency doubling the output of a dye laser pumped by the second harmonic of the Nd:YAG laser. The selection and recollimation of the VUV beam was achieved by the same procedure as described above for the excitation sequence 共2兲. The tunable UV radiation used in sequences 共1兲 and 共2兲 to induce transitions from the B( v ⬙ ⫽0,3,N ⬙ ⫽0) levels to the n ⬘ s and n ⬘ d Rydberg series located below the v ⫹ ⫽0, N ⫹ ⫽1 ground state of ortho H⫹ 2 was generated by doubling the frequency of a dye laser. This UV beam crossed the H2 supersonic jet and the VUV beam at right angles. At their intersection point the laser beams had diameters of ⬇1 mm to match the size of the skimmed supersonic jet. The dye laser wave numbers were calibrated by recording optogalvanic spectra of argon or neon and laser induced fluorescence spectra of molecular iodine and making reference to tabulated wave numbers.25,26 The bandwidth of the dye lasers amounted to 0.15 cm⫺1 but was reduced to 0.02 cm⫺1 when operated with intracavity etalon. Under these conditions, VUV radiation with a bandwidth of 0.1 cm⫺1 and UV radiation with a bandwidth of 0.04 cm⫺1 could be generated. The millimeter waves used in the last step of the excitation sequences were generated by a backward wave oscillator based synthesizer 共AMC Chemnitz兲 operated in a continuous mode. The frequencies were phase locked to a high harmonic of a microwave reference frequency. The specified spectral width of the millimeter waves is of the order of 1–2 kHz and the stability of the millimeter wave frequency is estimated to be better than 10⫺7 . The source can deliver up to 2 mW of broadly tunable radiation in the range 118 –184 GHz. After exiting the synthesizer via a horn, the millimeter wave beam was refocused by a parabolic mirror so that the beam traversed the excitation region at right angles to the VUV beam

and the gas jet and antiparallel to the UV beam. Two different horns were used over the whole tunable range to ensure that the waist of the millimeter wave beam did not exceed 1 cm in the photoexcitation region. An adjustable attenuator with a maximum attenuation of 60 dB was placed at the exit of the millimeter wave synthesizer and the power was carefully adjusted to avoid undesirable power broadening of the millimeter wave transitions. B. The photoexcitation region and the measurement procedure

Photoexcitation takes place between the second and the third plates of a set of five parallel resistively coupled cylindrical plates made of demagnetized stainless steel.15,16 The photoexcitation region was carefully designed to reduce stray electric and magnetic fields that can cause undesirable line shifts and line broadenings. The stray magnetic fields were reduced by more than two orders of magnitude by two concentric mumetal tubes placed around the photoexcitation region and the adjacent time-of-flight 共TOF兲 spectrometer. To reduce the effects of stray electric fields, the extraction plates surrounding the photoexcitation region were carefully polished and designed so that the minimum distance between the metallic surfaces and the photoexcitation region exceeds 15 mm. Moreover, the timing of the experimental cycles were synchronized with the 50 Hz frequency of the mains to eliminate uncontrollable stray fields originating from periodic fluctuations. Under these conditions, the stray electric fields remained stable over long periods. They were determined in regular intervals by measuring the Stark shift of reference transitions according to the procedure described in Ref. 27 and, if necessary, compensated to a level below 1 mV/cm by applying compensation voltages to the extraction plates. After concerns were expressed that stainless steel was a less adequate material than molybdenum for these types of experiments, an identical extraction region was made in molybdenum. However, the stray fields in the molybdenum setup turned out to be larger than in the stainless steel setup, presumably because the molybdenum plates are much harder to polish. The experimental chamber was evacuated by turbomolecular pumps and the background pressure during the experimental cycles amounted typically to 3⫻10⫺7 mbar. The resolution of better than 1 MHz of millimeter wave spectroscopy can only be fully exploited if the states that are probed by the millimeter wave transitions are long lived 共i.e., ␶ ⭓1 ␮ s). Prior to recording millimeter wave spectra, it is thus imperative to monitor the decay of the Rydberg states prepared by laser excitation and identify the long-lived Rydberg states. An almost complete picture of the decay dynamics of the Rydberg states could be derived from laser experiments in which signals originating from various decay mechanisms were collected. 共1兲 Autoionization. The autoionization and the direct ionization signals were detected by extracting the H⫹ 2 ions with a pulsed electric field applied after the laser pulses and monitoring the H⫹ 2 signal at a microchannel plate detector located at the end of the TOF tube.

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J. Chem. Phys., Vol. 121, No. 23, 15 December 2004

共2兲 Predissociation. Predissociation of the ns and nd Rydberg states of H2 leads to the formation of n⫽2 atomic fragments. When the predissociation is rapid enough that it takes place during the laser pulses, the UV radiation used to excite the Rydberg states also efficiently ionizes the excited H fragments. The predissociation can thus be detected by monitoring the H⫹ ion signal at the end of the TOF tube. 共3兲 Pulsed field ionization of long-lived Rydberg states. Long-lived Rydberg states remain as neutral species in the photoexcitation region until an electric field is applied that is large enough to cause their field ionization. Either the H⫹ 2 or the electron signal produced by pulsed field ionization can be detected, the electron detection scheme providing the advantage that the field-ionization signal can be distinguished from the autoionization or direct ionization signal. Indeed, unlike the H⫹ 2 ions, the electrons produced by photoionization during the laser pulse leave the measurement volume before the application of the field-ionization pulse and do not contaminate the field-ionization signal. Monitoring the magnitude of the field-ionization signal as a function of the time delay between photoexcitation and pulsed field ionization provides additional information on the lifetimes of the Rydberg states. After identification of the long-lived Rydberg states, millimeter wave spectra can be recorded. To obtain backgroundfree spectra, the magnitude of the pulsed electric field used to ionize the Rydberg states was adjusted so that it did not cause the ionization of the initial states of the millimeter wave transitions but efficiently ionized the final states 共other more sophisticated methods based on selective field ionization can also be used for that purpose16兲. The amplitude of the fast rising electric field pulse ranged from 44 V/cm to detect transitions in the lower part of the n range investigated (51d→54p/ f ) to 31 V/cm in the higher part (60d →64p/ f ). The time delay between the laser pulse and the field-ionization pulse restricts the time available for the measurement of the transition frequencies. Delay times of 1 ␮s, leading to linewidths of 1 MHz, were used when recording survey spectra. For the determination of precise transition frequencies, the field-ionization pulse was applied 3 ␮s after the laser pulse resulting in linewidths of about 300 kHz.

C. The error budget

The millimeter wave spectra provide highly precise transition frequencies between high Rydberg states but do not give the absolute positions of the Rydberg states with respect to the ground neutral state or to a specific ionization threshold with the same precision. The precision and accuracy of the positions with respect to the ground state remain limited by the precision and accuracy of the transition frequencies between the ground and Rydberg states, the accuracy of the positions with respect to a given ionization limit by the accuracy with which the Rydberg series can be extrapolated to n⫽⬁. A systematic procedure of building combination differences can be exploited to derive a network of relative positions of all Rydberg states connected by the observed transitions. The procedure followed in the present investigation was to arbitrarily set the position of one hyperfine structure

Rydberg states of molecular hydrogen

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component of the 51d Rydberg state, the 51d1 1 (G ⫹ ⫽1/2)F⫽0 component, to zero and refer all other positions to that of this component. Millimeter wave spectra were then measured, using excitation sequence 共2兲, from all hyperfine components of the 51d1 1 Rydberg state accessible within the laser bandwidth to all higher n p and n f Rydberg states accessible within the tuning range of the millimeter wave source, i.e., n⫽54 and n⫽55. Repeating the same procedure for the 52d1 1 states 共transitions to n⫽55 and 56兲, the 53d1 1 states 共transitions to n⫽56 and 57兲, etc., permitted to derive both initial and final state combination differences and hence to construct a set of relative positions of p, d, and f Rydberg states. Measurements of millimeter wave transitions from n p1 1 to ns and nd Rydberg states using sequence 共3兲 were finally used to extend the network of relative positions to ns1 1 and nd1 2 states in the range n⫽58– 62. This procedure leads to a certain redundancy in the data set which can be exploited to extract statistical information 共average values and standard deviations兲 on the transition frequencies and the combination differences. When establishing the error budget of the measurement, one needs to determine both the errors in the transition frequencies and the error inherent to the procedure used to derive the combination differences. It is well known to spectroscopists that an analysis based on combination differences does not represent the statistically optimal procedure and should be avoided whenever a fully satisfactory model can be used to describe both the initial and final state level structures.28 In the present case, the MQDT analysis of the gerade Rydberg states has not yet reached as satisfactory a level as that of the ungerade Rydberg states, and the disadvantages of an analysis of the transition frequencies using a model for both gerade and ungerade Rydberg states still outweigh the disadvantages of an analysis of the ungerade Rydberg states based on combination differences. The overall uncertainty of the experimental frequencies does not only originate from the calibration of the millimeter wave frequencies. Potential errors, such as AC Stark shifts, Doppler shifts caused by a slight misalignment of the H2 beam and the millimeter wave beam, Stark shifts caused by residual stray fields and pressure shifts, can easily affect the results of spectroscopic measurements on high Rydberg states and needed to be carefully minimized and included in the estimation of the experimental uncertainty. The various sources of systematic errors considered in the present study are now discussed in turn: 共1兲 Calibration errors. The uncertainty inherent to the millimeter wave frequency calibration is determined by the stability of the microwave reference frequency and can be safely estimated to be less than 50 kHz over the whole tuning range of the source on the basis of the specifications of the microwave reference generator and of test measurments of transitions between high Rydberg states of Ar 共Ref. 29兲 and Kr 共Ref. 27兲. 共2兲 AC Stark shifts. Transitions between neighboring Rydberg states are characterized by unusually large transition moments and are very easily broadened and shifted by the AC Stark effect induced by long wavelength radiation.30 To

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minimize the error resulting from AC Stark shifts by the millimeter wave radiation, each line was recorded at the minimal power compatible with a satisfactory signal-to-noise ratio. For the strongest lines, an attenuation of 40 dB was necessary; for the weakest, 20 dB was sufficient. In addition, it was verified that a further decrease of the millimeter wave power did not lead to an alteration of the resonance frequencies. The contribution of the AC Stark shifts to the transition frequencies could thus be reduced well below the experimental linewidths and is estimated to be less than 50 kHz. 共3兲 Doppler shifts. Deviations of the angle between the millimeter wave and molecular beam propagation axes from 90° lead to Doppler shifts. To quantify the Doppler shifts in our measurements, experiments were carried out using neat H2 共mean velocity in the supersonic expansion of ⬇2500 m/s) and a 50:1 Kr:H2 mixture 共mean velocity of 450 m/s兲. No changes in the central frequencies of the resonances could be detected in these measurements indicating that the contribution of Doppler shifts to the transition frequencies is much less than the experimental width. The error was thus estimated to be less than 50 kHz. 共4兲 Pressure shifts. Pressure shifts are known to be particularly pronounced in high Rydberg states30–33 and need to be considered when determining ionization potentials by extrapolation of Rydberg series. The contributions to the pressure shift are the stabilization 共redshift兲 resulting from the ion-induced dipole interaction between the ionic core of the Rydberg state and the neutral particles located within the Rydberg electron orbit, and the collisional phase shift of the Rydberg electron with these particles which can lead to a displacement to higher 共blueshift兲 or lower 共redshift兲 energies.32 The pressure shift becomes noticeable as soon as the Rydberg electron orbit becomes similar to the average distance from the ion core to the nearest neutral foreign atom or molecule. Within a Rydberg series, the pressure shift is thus negligible at low n values, increases at intermediate n values to a value characteristic of the neutral foreign particles and their densities, and remains constant at high n values. The pressure shift at high n values is also to a good approximation ᐉ independent. Pressure shifts of the Rydberg states energies induced by rare gases and H2 molecules at the local pressure of the supersonic expansion can be estimated to be of the order of several MHz under our experimental conditions and are approximately the same for the initial and final states of the millimeter wave transitions. The millimeter wave experiments, however, are only sensitive to the differential pressure shifts, i.e., to the differences of pressure shifts between upper and lower Rydberg states connected by the millimeter wave transitions. These differential pressure shifts are much smaller than the absolute shifts. To quantify the pressure shifts in our measurements, experiments were carried out using different carrier gases 共neat H2 , Kr兲 and using different nozzle stagnation pressures 共between 1 and 6 bar兲. Unfortunately, no measurable pressure shifts could be detected in any of these experiments and we can only conclude that, at the values of n between 50 and 65 of interest in the present study, the pressure shifts are sufficiently n and ᐉ independent so that the shifts in the upper and lower Rydberg states cancel out in the transition frequency. The error caused

Osterwalder et al.

by the pressure shift could thus also be estimated to be less than 50 kHz. 共5兲 Stark shifts. Measurements of stray electric fields using reference transitions according to the procedure described in Ref. 27 were carried out in regular intervals to ensure that the stray fields never exceeded 3 mV/cm. In the range of n values probed experimentally (n⬇60) the half width of the linear Stark manifold at a field of 3 mV/cm amounts to ⬇7⫻10⫺4 cm⫺1 . The spacing between adjacent Rydberg states being about 1 cm⫺1 at n⫽60, only Rydberg states with quantum defects of less than ⬇7⫻10⫺4 are subject to a linear Stark effect under these conditions. Even the least penetrating states of interest here, the f states, have quantum defects of more than 10⫺2 共see Sec. III and Fig. 16兲. The s, p, d, and f Rydberg states observed experimentally are therefore expected to be quadratically shifted by the stray fields. The Stark shifts at fields of 3 mV/cm could not be measured directly but were estimated from a measurement of the Stark shift of the 51d→55p/ f transitions in a field of 40 mV/cm followed by quadratic downscaling to a field of 3 mV/cm. From measured shifts of ⬇25 MHz for the d- f transitions and of less than 5 MHz for the d-p transitions at 40 mV/cm, maximal Stark shifts of ⬇30 kHz for the d-p transitions and of ⬇140 kHz for the d- f transitions at n⫽55 at 3 mV/cm can be estimated. Considering the n 7 scaling of the polarizability of the Rydberg states, these shifts increase to about 90 kHz for the n p, and 400 kHz for the n f Rydberg states at the highest n values probed experimentally (n ⫽64). The various contributions to the experimental uncertainties in the transition frequencies are summarized in Table II and result in overall uncertainties originating from systematic errors varying from 105 kHz and 185 kHz for the transitions to n p and n f states at n⫽55, respectively, to about 180 kHz and 410 kHz at n⫽64. Statistical analysis of repeated measurements of selected transitions in several distinct experimental sessions spread over a period of more than 2 years revealed that the maximum overall uncertainty estimated above lies a factor of about 2 above the experimental standard deviations, but also clearly revealed the larger uncertainties in the frequencies of transitions to f states compared to p states and the increase of the uncertainties with growing n values. The error budget is dominated by the contribution of the Stark shifts, and future experiments aimed at a still higher accuracy will need to pay more attention to the stray field minimization procedure. A reduction of the stray field well below 1 mV/cm is possible;27 however, keeping the stray field at that level during measurement periods of several months is demanding and slows down the data acquisition. The increase of the experimental uncertainties with growing n values motivated our choice of building the combination differences starting at the lowest n values. In this way, the error accumulated in the successive steps needed to construct the relative energy map is paralleled by the gradual increase of the uncertainties in the transition frequencies. As a result, the uncertainties in the relative positions of the different p and f levels only rarely exceed three times the estimated systematic errors in the transition frequencies connect-

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TABLE II. Error budget in the determination of the millimeter wave transition frequencies. Estimated maximal error 共kHz兲

Method of estimation

Frequency calibration AC Stark shift Doppler shift Pressure shift Stark shift 共55p兲 Stark shift 共55f兲

⭐50 ⭐50 ⭐50 ⭐50 ⭐30a ⬇140b

Specification of microwave reference frequency Attenuation of millimeter wave power Change of carrier gas Change of carrier gas and variation of pressure Application of dc fields Application of dc fields

Overall uncertainty 55p 共64p兲 Overall uncertainty 55f 共64f兲

⭐105 共135兲 180 共410兲

Error source

a

This value increases to 90 kHz at n⫽64. This value increases to 400 kHz at n⫽64.

b

ing these levels to the rest of the energy map 共see Table VI below兲.

in Sec. VII which include all spins but for a discussion of the overall level structures the hyperfine structure can be disregarded.

III. GENERAL STRUCTURE OF p AND f RYDBERG MANIFOLDS

A. Level structure of p states without spin

We consider in this section the general structure of p and f Rydberg levels of positive total parity of ortho H2 (I ⫽1). Equations 共1兲–共3兲 show that the experimental excitation sequences limit the possible values of ᐉ to 0–3, of N to 0–2, and of F to 0–3. We begin with a qualitative discussion of the channel structures and expected energy level patterns for p and f states by disregarding all spins, and we shall discuss the complications arising from the presence of the spins subsequently. Figure 1 is an energy level diagram of the observed and theoretical structures associated with the p and f electrons for n⫽55 which we take as an example here. The level patterns occurring for the other n values studied in this work are very similar. The calculated levels shown in Fig. 1 actually correspond to the final calculations described

FIG. 1. Observed and calculated p and f energy levels 共positive total parity兲 near n⫽55 associated with the v ⫹ ⫽0,N ⫹ ⫽1 ionization threshold of ortho H2 and plotted for different values of the total angular momentum F. Calculated p (ᐉ⫽1) levels are enclosed in boxes with the approximate BornOppenheimer case 共b兲 character indicated. The remaining levels correspond to ᐉ⫽3. The dashed line indicates the position of the n⫽55 level of zero quantum defect and is drawn at the position ⫺RyH2 /552 below the G ⫹ ⫽1/2 hyperfine component of the N ⫹ ⫽1 ionic level.

The channel structure of the p Rydberg states in H2 is well known.33 In Hund’s case 共b兲, each channel is characterized by the values of ⌳ (⌳⫽0 or 1兲 and N, by the total parity (⫺1) p (p even or odd兲, and by the orbital angular momentum ᐉ of the Rydberg electron. In Hund’s case 共d兲— which is the case of primary interest for the high levels discussed here—once ᐉ uncoupling is complete, ⌳ is no longer a good quantum number but is replaced as good quantum number by the quantum number N ⫹ associatied with the core rotation. The parity quantum number p in this limit simply equals N ⫹ ⫹ᐉ because the core is in a 2 ⌺ ⫹ g state with levels ⫹ of parity (⫺1) N . The unitary transformation which connects the Hund’s case 共b兲 to the Hund’s case 共d兲 states has elements 具 ⌳ 兩 N ⫹ 典 (Npᐉ) , where the symbols in the bracket are the quantum numbers which are specific to each case, and the symbols written as superscripts are quantum numbers common to both cases. The transformation matrix has dimension 2ᐉ⫹1 (⫽3 for ᐉ⫽1 states兲, but, for a given parity p, it reduces to dimension ᐉ⫹1 or ᐉ 共or less when N⬍ᐉ), owing to the fact that for a given N value the core may rotate with N ⫹ ⫽N⫺ᐉ, N⫺ᐉ⫹1,..., N⫹ᐉ. Depending on the values of p and ᐉ only even or odd values of N ⫹ give the correct total parity. For N⫽0, only the single body-frame 关case 共b兲兴 channel ⌳⫽0 exists so that the case 共b兲 (⌳⫽0) and case 共d兲 (N ⫹ ⫽1) representations coincide and the transformation consists of the single element 具 ⌳⫽0 兩 N ⫹ ⫽1 典 (N⫽0,p⫽0,ᐉ⫽1) ⫽1. Consequently, even at n⬇55, the N⫽0 levels retain a pure ⌳ ⫽0 character, and their positions can be estimated directly with the ⌳⫽0 body-frame quantum defects. Near the equilibrium internuclear separation (R⬇2a 0 ), the quantum de3 ⫹ fects amount to 0.18 for the 1 ⌺ ⫹ u p and to 0.45 for the ⌺ u p channels 共see Fig. 11 below兲, so that at n⫽55 共where the Rydberg interval is 1.3 cm⫺1 ) the 1 ⌺ ⫹ u N⫽0 level is expected to lie 0.18⫻1.3 cm⫺1 ⫽0.23 cm⫺1 below the position of 124 439.447 cm⫺1 of the n⫽55 level of zero quantum defect associated with the v ⫹ ⫽0,N ⫹ ⫽1 threshold. SimiN⫽0 state is expected to lie 0.45 larly, the 3 ⌺ ⫹ u

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⫻1.3 cm⫺1 ⫽0.59 cm⫺1 below the n⫽55 level of zero quantum defect. These considerations are illustrated by Fig. 1 where the calculated p manifolds of states for n⫽55 are highlighted by rectangular frames, and the level of zero quantum defect is indicated by a horizontal dashed line. In Fig. 1 each frame stands for a single spinless level and the spin hyperfine structure inside each frame should be ignored at this point. The separate treatment of singlet and triplet states is justified by the fact that the singlet-triplet splitting amounts to (0.45⫺0.18)⫻1.3 cm⫺1 ⫽0.35 cm⫺1 and is still an order of magnitude larger than the largest hyperfine splitting in the ion core (0.043 cm⫺1 ), so that singlet and triplet states reach Hund’s case 共d兲 coupling separately, as already pointed out in the Introduction. For N⫽1, one also encounters a one-channel situation because p⌳ states with N⫽1 and positive total parity (p even兲 are of ⌸ ⫺ symmetry, and therefore do not interact with states of ⌸ ⫹ or ⌺ ⫹ symmetry. In Hund’s case 共d兲, N ⫹ takes the values 0, 1, and 2, but only the N ⫹ ⫽1 states have the correct positive total parity. Consequently, the transformation consists of a single element 具 ⌳⫽1 兩 N ⫹ ⫽1 典 (N⫽1,p⫽0,ᐉ⫽1) ⫽1, and the N⫽1 levels retain pure ⌸ ⫺ character for all n. The quantum defect functions at R⬇2a 0 共see Fig. 11兲 imply that the singlet p levels with N⫽1 must lie 1.3 ⫻0.08 cm⫺1 ⫽0.10 cm⫺1 above, and their triplet counterparts 1.3⫻0.06 cm⫺1 ⫽0.08 cm⫺1 below the n⫽55 level of zero quantum defect, as illustrated by Fig. 1. For N⫽2, a two-channel situation arises because both the ⌺ ⫹ and ⌸ ⫹ Hund’s case 共b兲 components possess the required positive total parity, and both N ⫹ ⫽1 and N ⫹ ⫽3 case 共d兲 components are compatible with it. These case 共d兲 levels have effective quantum defects that are nearly 1:1 mixtures of the ⌳⫽0 and ⌳⫽1 body-frame quantum defects 共precisely 4/9:5/9 for N ⫹ ⫽1), and therefore lie roughly midway between the N⫽0 and N⫽1 level positions discussed above 共see Fig. 1兲. As can be seen in Fig. 1, all ᐉ⫽1 levels observed for n⫽55—with a single exception—correspond to N⫽1. The reason for this behavior lies in the predissociation of the N ⫽0 and N⫽2 levels which, unlike the N⫽1 levels that are pure ⌸ ⫺ levels, can interact with the dissociative ⌺ ⫹ u state 共see Sec. IV A below兲. B. Level structure of f states without spin

The f states are nonpenetrating states characterized by small quantum defects, and therefore their positions must be situated near the levels of zero quantum defect as illustrated for n⫽55 on the right-hand side of Fig. 1 where the f levels are situated outside the frames. These levels are displayed in Fig. 2 on a tenfold increased energy scale. It is known from the study of the 4 f complexes of levels in H2 that the singlettriplet separations are very small in these nonpenetrating states because the exchange interaction is almost negligible. From a singlet-triplet separation of the order of 0.1 cm⫺1 at n⫽4,34 a splitting of about 0.000 04 cm⫺1 can be inferred at n⫽55 by use of the (n * ) ⫺3 scaling law. The singlet-triplet splitting is thus vanishingly small on the scale of Fig. 2 and is dwarfed by the hyperfine splitting of the ion core which is

Osterwalder et al.

FIG. 2. Tenfold enlarged view 共compared to Fig. 1兲 of the observed and calculated f energy levels 共positive total parity兲 near n⫽55 associated with the v ⫹ ⫽0,N ⫹ ⫽1 ionization threshold of ortho H2 . The dotted lines indicate the energetic positions ⫺RyH2 /552 below the G ⫹ ⫽1/2 and G ⫹ ⫽3/2 hyperfine components of the N ⫹ ⫽1 ionic level. The full hyperfine structure ⫹ ⫹ 2 of H⫹ 2 ( v ⫽0,N ⫽1) displaced by ⫺RyH2 /55 is shown on the right-hand side.

three orders of magnitude larger. One therefore expects separate case 共e兲 n f series to converge to each of the two groups ⫺1 of H⫹ for N ⫹ 2 hyperfine levels separated by ⬇0.043 cm ⫽1 and corresponding to the two total core spin values G ⫹ ⫽1/2 and G ⫹ ⫽3/2. The two G ⫹ core-spin levels of zero quantum defect at n⫽55 are indicated in Fig. 2 by dashed lines, and the groups of 55f levels associated with each of them can easily be identified. None of these levels is separated from its nearest level of zero quantum defect by more than 0.020 cm⫺1 . Dividing this maximum deviation by the Rydberg interval 1.3 cm⫺1 , an upper bound of 0.015 can be derived for the body-frame quantum defects, which indeed agrees with the largest body-frame quantum defect represented in Fig. 16 near R⫽2a 0 (⌳⫽0, ␩ ⫽0.015). If the hyperfine structure visible in Fig. 2 is disregarded, the levels of zero quantum defect associated with each of the two n⫽55 states form three groups corresponding to ᐉ⫽3, N ⫹ ⫽1, and N⫽2, 3, and 4. The pattern is identical 关but downscaled by a factor (4/55) 3 ⬃4⫻10⫺4 ] to that reported by Herzberg and Jungen35 for the 4 f states 共see their Fig. 3兲. Although the couplings of states are more complicated in the ᐉ⫽3 case, the basic structure is qualitatively similar to that discussed above for the N⫽0, 1, and 2 components of the p levels at each value of S; the N⫽2 level lies lowest in energy, the N⫽3 level highest, and the N⫽4 level in between. We see finally that all but two levels observed experimentally correspond to N⫽2, the only rotational quantum number accessible when the spins are neglected. C. Structure of p and f Rydberg manifolds considering electron and nuclear spin

Having acquired a qualitative understanding of the overall level structure expected for a given n value, we now turn to the discussion of the hyperfine structure. The first task consists of determining the number of spin-rotational chan-

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J. Chem. Phys., Vol. 121, No. 23, 15 December 2004

Rydberg states of molecular hydrogen

and 1 and ⌳⭐ᐉ which possess such J rotational levels. The same procedure for ᐉ⫽3 yields the 37 f Born-Oppenheimer or body-frame channels listed in Table IV. The number of channels can also be obtained by starting from the uncoupled 共non-Born-Oppenheimer兲 case 共e兲 关i.e., case 共d兲 with spins兴 limit. We take the same example, start again with F⫽3 and p⫽0, and subtract the spin-orbital angular momentum j of the Rydberg electron ( j takes the values ᐉ⫾1/2, e.g., j⫽1/2 and 3/2 for ᐉ⫽1) from F to obtain F⫹ , the total angular momentum of the residual ion core. The total core spin quantum number G ⫹ can take the values G ⫹ ⫽1/2 and 3/2. By subtracting G⫹ from F⫹ , we obtain N⫹ , the core rotational angular momentum exclusive of the spins. Since the Rydberg orbital angular momentum ᐉ is odd in the levels studied here, only odd values of N ⫹ must be retained to preserve the required total parity. Tables III and IV give the results for ᐉ⫽1 and 3. The same numbers of channels, 18 and 37 for ᐉ⫽1 and 3, respectively, are obtained as above for case 共a兲, as is required physically. The hyperfine coupling schemes which implicitly have been assumed in the above discussion are (a␤ J ) and (e关 b ␤ S ⫹ 兴 ), respectively. The particular notation used here for case 共e兲 expresses the fact the electron spin-orbital angular momentum j couples to the ion core which itself approaches case (b␤ S ⫹ ) 共see Sec. I兲. Each set of channels listed in Tables III and IV corresponds to a single limiting case. The actual level positions result from the interaction between all the channels of the same F and p values—including interactions between the p and f channel manifolds and vibrational channel interactions, as will be discussed below. Nevertheless, the levels studied in this work near the v ⫹ ⫽0, N ⫹ ⫽1 hyperfine thresholds

TABLE III. Hyperfine Rydberg p channels in ortho H2 (I⫽1): F⫽3, ⫹ total parity. The symmetry labels e and f correspond to the definition given on p. XI of Ref. 44. Eighteen body-frame channels J,S,⌳,⍀ Hund’s case (a␤ J ) ⌳(⍀ ⫾ )⫽ 1 ⌺ ⫹ (0 ⫹ ), 1 ⌸(1 ⫹ ), 3 ⌺ ⫹ (1 ⫹ ), 3 ⌸(0 ⫹ ,1⫹ ,2⫹ ) f 2S⫹1 ⫾ 1 J⫽3 ⌳(⍀ )⫽ ⌸(1 ⫺ ), 3 ⌺ ⫹ (0 ⫺ ,1⫺ ), 3 ⌸(0 ⫺ ,1⫺ ,2⫺ ) e 2S⫹1 ⫾ 1 ⫹ J⫽4 ⌳(⍀ )⫽ ⌺ (0 ⫹ ), 1 ⌸(1 ⫹ ), 3 ⌺ ⫹ (1 ⫹ ), 3 ⌸(0 ⫹ ,1⫹ ,2⫹ )

F⫽3,⫹total parity J⫽2 e

2S⫹1

Eighteen laboratory-frame channels N ⫹ ,F ⫹ ,G ⫹ , j Hund’s case (e关 b␤ S ⫹ 兴 ) F⫽3,⫹total parity j⫽ 1 , F ⫹ ⫽ 5 N ⫹ (G ⫹ )⫽1( 3 ),3( 1 , 3 ) 2 2 2 2 2 7

1 3

3

F ⫹ ⫽ 2 N ⫹ (G ⫹ )⫽3( 2 , 2 ),5( 2 ) 3 j⫽ 2 ,

3 F ⫹⫽ 2 5 F ⫹⫽ 2 7 ⫹ F ⫽2 9 ⫹ F ⫽2

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1 3 3 N ⫹ (G ⫹ )⫽1( 2 , 2 ),3( 2 ) 3 1 3 N ⫹ (G ⫹ )⫽1( 2 ),3( 2 , 2 ) 1 3 3 ⫹ ⫹ N (G )⫽3( 2 , 2 ),5( 2 ) 3 1 3 ⫹ ⫹ N (G )⫽3( 2 ),5( 2 , 2 )

nels and/or states that arise for given total angular momentum quantum number F and parity, which are the only good quantum numbers in field-free space. Starting, for example, with F⫽3, positive parity (p ⫽0) and I⫽1, and adopting the Born-Oppenheimer 关bodyframe case 共a兲兴 limit, we first subtract the nuclear spin from F to get J, the total angular momentum quantum number exclusive of nuclear spin. The possible values of J are 2, 3, and 4 and the corresponding levels must be of e, f , and e parity, respectively, in order to preserve p⫽0. Eighteen p channels, listed in Table III, are obtained by identifying all the 2S⫹1 ⌳ ⍀ ⫾ electronic states with total electron spin S⫽0

TABLE IV. Hyperfine Rydberg f channels in ortho H2 (I⫽1): F⫽3, ⫹ total parity. Thirty-seven body-frame channels J,S,⌳,⍀ Hund’s case (a␤ J ) F⫽3, ⫹ total parity

⌳(⍀ ⫾ )⫽ 1 ⌺ ⫹ (0 ⫹ ), 1 ⌸(1 ⫹ ), 1 ⌬(2 ⫹ ), 3 ⌺ ⫹ (1 ⫹ ), 3 ⌸(0 ⫹ ,1⫹ ,2⫹ ), 3 ⌬(1 ⫹ ,2⫹ ), 3 ⌽(2 ⫹ ) 2S⫹1 ⌳(⍀ ⫾ )⫽ 1 ⌸(1 ⫺ ), 1 ⌬(2 ⫺ ), 1 ⌽(3 ⫺ ), 3 ⌺ ⫹ (0 ⫺ ,1⫺ ), 3 ⌸(0 ⫺ ,1⫺ ,2⫺ ), 3 ⌬(1 ⫺ ,2⫺ ,3⫺ ), 3 ⌽(2 ⫺ ,3⫺ ) 2S⫹1 ⌳(⍀ ⫾ )⫽ 1 ⌺ ⫹ (0 ⫹ ), 1 ⌸(1 ⫹ ), 1 ⌬(2 ⫹ ), 1 ⌽(3 ⫹ ), 3 ⫹ ⌺ (1 ⫹ ), 3 ⌸(0 ⫹ ,1⫹ ,2⫹ ), 3 ⌬(1 ⫹ ,2⫹ ,3⫹ ), 3 ⌽(2 ⫹ ,3⫹ ,4⫹ )

J⫽2 e

2S⫹1

J⫽3 f J⫽4 e

Thirty-seven laboratory frame channels N ⫹ ,F ⫹ ,G ⫹ , j Hund’s case (e关 b␤ S ⫹ 兴 ) F⫽3, ⫹ total parity

5

1

j⫽ 2 , F ⫹ ⫽ 2

3 F ⫽2 5 ⫹ F ⫽2 7 F ⫹⫽ 2 9 F ⫹⫽ 2 11 F ⫹⫽ 2 7 1 ⫹ j⫽ 2 , F ⫽ 2 3 F ⫹⫽ 2 5 F ⫹⫽ 2 7 F ⫹⫽ 2 9 F ⫹⫽ 2 11 ⫹ F ⫽ 2 13 ⫹ F ⫽ 2 ⫹

1 3

N ⫹ (G ⫹ )⫽1( 2 , 2 ) ⫹



1 3

3

N (G )⫽1( 2 , 2 ),3( 2 ) 3 1 3 N ⫹ (G ⫹ )⫽1( 2 ),3( 2 , 2 ) 1 3 3 N ⫹ (G ⫹ )⫽3( 2 , 2 ),5( 2 ) 3 1 3 ⫹ ⫹ N (G )⫽3( 2 ),5( 2 , 2 ) 1 3 3 N ⫹ (G ⫹ )⫽5( 2 , 2 ),7( 2 ) 1 3 N ⫹ (G ⫹ )⫽1( 2 , 2 ) 1 3 3 N ⫹ (G ⫹ )⫽1( 2 , 2 ),3( 2 ) 3 1 3 N ⫹ (G ⫹ )⫽1( 2 ),3( 2 , 2 ) 1 3 3 ⫹ ⫹ N (G )⫽3( 2 , 2 ),5( 2 ) 3 1 3 ⫹ ⫹ N (G )⫽3( 2 ),5( 2 , 2 ) 1 3 3 N ⫹ (G ⫹ )⫽5( 2 , 2 ),7( 2 ) 3 1 3 N ⫹ (G ⫹ )⫽5( 2 ),7( 2 , 2 )

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contain a dominant N ⫹ ⫽1 contribution, so that Tables III and IV 共or analogous Tables for F⫽3) can be used to estimate the number of levels of given F and p values in any given Rydberg interval. For example, Table III indicates that four of the 18 channels with ᐉ⫽1 have N ⫹ ⫽1, and Fig. 1 indeed contains four calculated p levels with F⫽3. Three of these levels have N⫽2 and one has N⫽1 共the only one observed兲. Similarly, Table IV predicts that, for ᐉ⫽3, ten levels with F⫽3 and N ⫹ ⫽1 should occur, four having G ⫹ ⫽1/2 and six having G ⫹ ⫽3/2. These can indeed be found on the right-hand side of Fig. 2 共the N⫽3 levels with G ⫹ ⫽1/2 and G ⫹ ⫽3/2 actually are doublets with a splitting too small to be seen on the scale of the figure兲. D. Case „a… to case „e… frame transformations

Although the levels observed in this work are nearer to cases 共e兲 or 共d兲 than to cases 共a兲 or 共b兲, it is important to work out the frame transformations, in other words the unitary matrices that connect the various coupling cases. Indeed, molecular multichannel quantum defect theory rests on the use of these transformations. They allow coupled multichannel electron-core wave functions to be set up that take the form of superpositions of Born-Oppenheimer products at short range 关case (a␤ J ) in our case兴, and become asymptotically superpositions of decoupled core-electron scattering functions 共case 关 e关 b␤ S ⫹ 兴 ) in our case兲 as they are appropriate for use in the framework of collision theory. In this way, nonadiabatic effects 共the breakdown of the BornOppenheimer factorization兲 are included in an elegant way, and many intermediate situations can be accounted for 共see, e.g., Ref. 36 or various papers reprinted in Ref. 37兲. We have seen in Sec. III A that in the spinless case, and for a ⌺ ⫹ core 共which was implicitly assumed兲, the elements of the unitary transformation connecting Hund’s cases 共b兲 and 共d兲 are conveniently denoted as 具 ⌳ 兩 N ⫹ 典 (Npᐉ) . Tables III and IV suggest an analogous notation for the transformation connecting the electron-nuclear spin representations (a␤ J ) ⫹ and (e关 b␤ S ⫹ 兴 ), namely, 具 S⌳⍀J 兩 N ⫹ G ⫹ F ⫹ j 典 (Fp,S I,ᐉs) , where the quantum numbers common to both limiting cases are listed as superscripts, and the bras and kets contain the quantum numbers which are good only in one limiting case 共see Table I for the definition of the quantum numbers兲. A total of ten quantum numbers need to be considered, four of which change between case 共a兲 and case 共e兲, as compared to four and one, respectively, in the spinless case. The dimension of the matrices is correspondingly increased: for ᐉ⫽3, N⫽3 without spin the dimensions are 4⫻4 or 3⫻3 depending on the parity, while in the case of ᐉ⫽3, F⫽3 described by Table IV a 37⫻37 transformation matrix has to be set up. Jungen and Raseev38 have presented a general discussion of Hund’s cases 共a兲, 共b兲, 共d兲, and 共e兲 in diatomic molecules with and without electron spin, and they have derived the expressions for the transformations between them, however, without taking nuclear spins into account. In Appendix A, their expressions are generalized to include the nuclear spins. This generalization is achieved in the following way: The most general expression of Ref. 38 connects Hund’s cases 共a兲 and 共e兲 关their Eq. 共12兲兴. By addition of the nuclear spin I, this

expression can be extended to connect the nuclear spin case (a␤ J ) to an intermediate coupling case ˜e that was introduced to facilitate the derivation of the overall transformation and that differs from case (e关 b␤ S ⫹ 兴 ) solely by the reversed ordering of the coupling of j and I. It is then sufficient to work out the transformations 具˜e兩 e关 a␤ J ⫹ 兴 典 and 具 e关 a␤ J ⫹ 兴 兩 e关 b␤ S ⫹ 兴 典 and multiply the matrices in order to obtain the desired transformation, as explained in Appendix A. The derivation given in Appendix A is in fact somewhat more general than the discussion in the present section because it allows for the possibility that the core can have ⌳ ⫹ ⬎0 共⌸ or ⌬ core兲, or that it can be in a ⌺ ⫺ state. Case (e关 b␤ S ⫹ 兴 ) is the appropriate case 共e兲 limit to be considered here because the H⫹ 2 ion core, with I coupled to S⫹ to yield G⫹ , is quite close to case (b␤ S ⫹ ), which by addition of the Rydberg electron yields (e关 b␤ S ⫹ 兴 ). Obviously, it is also possible to derive expressions for other case 共e兲 cases where the ion core is in a different hyperfine coupling case. One must also consider that H⫹ 2 is not exactly in case (b␤ S ⫹ ). As a result, the frame transformation matrix 具 a␤ J 兩˜e典具˜e兩 e关 a␤ J ⫹ 兴 典具 e关 a␤ J ⫹ 兴 兩 e关 b␤ S ⫹ 兴 典 given in Appendix A must be multiplied by the eigenvector matrix 共derived in Sec. V and Appendix B兲 transforming the b␤ S ⫹ basis functions into the ion eigenstates to get the exact spin-rotation electronic 共sre兲 frame transformation for the present problem

具 ␣ (sre) 兩 i (sre) 典 ⬅ 具 a␤ J 兩˜e典具˜e兩 e关 a␤ J ⫹ 兴 典具 e关 a␤ J ⫹ 兴 兩 e关 b␤ S ⫹ 兴 典 ⫻具 e关 b␤ S ⫹ 兴 兩 eexact典 ,

共4兲

where summation over repeated indices is implied. The last transformation in Eq. 共4兲 involves the expansion of the ion eigenfunction in the (b␤ S ⫹ ) basis. The (a␤ J ) representation is appropriate for the description of the electron motion in a small volume near the core, where the Rydberg electron ‘‘acquires’’ the quantum defect depending on its 2S⫹1 ⌳ ⍀ state and the momentaneous distance R between the nuclei. This representation allows the high-n Rydberg states to be linked to clamped-nuclei potential energy curves of low-n states. IV. OVERVIEW OF EXPERIMENTAL RESULTS AND SPECTRAL ASSIGNMENTS A. Laser spectra and their assignment

Low-resolution (0.3 cm⫺1 ) overview spectra of the Rydberg series corresponding to gerade upper levels of ortho H2 in the vicinity of the v ⫹ ⫽0, N ⫹ ⫽1 threshold that can be excited via the B ( v ⬙ ⫽0, J ⬙ ⫽0 and 2兲 levels are displayed in Figs. 3共a兲 and 3共b兲, respectively. The spectra were recorded without intracavity etalons in the dye lasers. In each case, the upper trace corresponds to the spectrum obtained by monitoring the H⫹ 2 yield and the lower trace to the predissociation spectrum recorded by measuring the H⫹ signal. As explained in Sec. II B, the H⫹ 2 signal contains contributions from direct ionization, autoionization, and pulsed field ionization. For energetic reasons, however, the H⫹ 2 signal below the N ⫹ ⫽1 threshold originates exclusively from the field ionization of long-lived Rydberg states. Two types of resonances can be distinguished in Fig. 3: The first type forms extended progressions which can be as-

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J. Chem. Phys., Vol. 121, No. 23, 15 December 2004

FIG. 3. Survey of the ns and nd Rydberg series of ortho H2 in the vicinity ⫹ ⫹ of the X 2 ⌺ ⫹ g ( v ⫽0,N ⫽1) ionization threshold. The spectra were recorded in a (1⫹1 ⬘ ) two-photon excitation sequence using 共a兲 the R(1) and 1 ⫹ 共b兲 the P(1) line of the X 1 ⌺ ⫹ g ( v ⫽0)⫺B ⌺ u ( v ⫽0) transition in the first excitation step. The positive trace corresponds in each case to the H⫹ 2 ions produced by pulsed field ionization or autoionization and the negative trace to the predissociation signal corresponding to ionization from the excited n⫽2 H fragment. A pulsed field of 350 V/cm was used for the field ionization and the extraction of the ions. The zero point of the wave number scale corresponds to the position of the X 1 ⌺ ⫹ g ( v ⫽0,J⫽0) ground neutral state.

signed to Rydberg series converging on the v ⫹ ⫽0, N ⫹ ⫽1,3 thresholds. The second type corresponds to strong resonances of variable widths and intensities which can be attributed to low-n Rydberg states associated with a vibra17,39,40 Both the tionally excited ( v ⫹ ⫽1 and 2兲 H⫹ 2 ion core. low-n and the vibrationally excited character of these interloper Rydberg states lead to their rapid decay by predissociation and, above the N ⫹ ⫽1 limit, also by autoionization. Because of interactions with these interloper states, the series converging on the v ⫹ ⫽0, N ⫹ ⫽1 and 3 limits show marked perturbations that manifest themselves either by sharp intensity changes or enhanced predissociation signals. Only three Rydberg series with a v ⫹ ⫽0 core, the ns1 1 , nd1 1 , and nd3 1 series 共the notation nᐉN ⫹ N is used here and in the following to label the Rydberg series, see Sec. I兲, carry intensity when excitation occurs via the N ⬙ ⫽0 level of the B state. All three series contribute to the spectra displayed in Fig. 3共b兲. The members of the ns1 1 and nd1 1 series decay by rapid predissociation ( ␶ ⭐5 ns) below n⫽30, but appear to be long lived above n⫽30. The nd3 1 series is very irregular and gains intensity in the range n⫽23– 30 关around 124 630 cm⫺1 in Fig. 3共b兲兴 by interaction with a low-n interloper state. The spectral features observed in Fig. 3共b兲 are also present in Fig. 3共a兲 but only represent a minor contribution to the intensity distribution. The spectra recorded via the B( v ⬙ ⫽0,N ⬙ ⫽2) level are dominated by the nd3 2 and nd3 3 series which also show pronounced perturbations in the vicinity of low-n interlopers. Rapid predissociation and autoionization of most Rydberg states observed between the N ⫹

Rydberg states of molecular hydrogen

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FIG. 4. Decay dynamics of the gerade Rydberg states of ortho H2 located ⫹ ⫹ below the X 2 ⌺ ⫹ g ( v ⫽0,N ⫽1) ionization threshold. Traces 共a兲, 共b兲, 共c兲, and 共d兲 were measured by detecting the pulsed field ionization signal 10 ns, 40 ns, 200 ns, and 1 ␮s after photoexcitation, respectively. The intensities of the spectra have been normalized to that of the transition to the 47d1 1 state. The field ionization was induced by a pulsed field of 350 V/cm amplitude. The zero point of the wave number scale corresponds to the position of the X 1⌺ ⫹ g ( v ⫽0,J⫽0) ground neutral state.

⫽1 and 3 ionization thresholds render these states unattractive for millimeter wave spectroscopic experiments. Longlived Rydberg states are primarily located below the v ⫹ ⫽0, N ⫹ ⫽1 limit between 124 400 and 124 450 cm⫺1 above the v ⵮ ⫽0, N ⵮ ⫽0 neutral ground state and are more conveniently accessed via the N ⬙ ⫽0 level of the B state. The results of an investigation of the decay dynamics of the Rydberg states in this region are displayed in Fig. 4, which shows a comparison between spectra obtained by applying a 350 V/cm field ionization pulse at delay times of ⬇10 ns 共a兲, 40 ns 共b兲, 200 ns 共c兲, and 1 ␮s 共d兲 following the laser pulse. At the shortest delay time 关trace 共a兲兴, the overlap of the rising edge of the electric field pulse with the laser pulse led to a slight Stark broadening of the resonances. The nd1 1 and ns1 1 Rydberg states can be identified on the basis of their known quantum defects. Most members of the strong nd1 1 and the weaker ns1 1 series in the range n⫽42– 60 have lifetimes of 1 ␮s or more. A local perturbation caused by the 18d3 1 Rydberg state leads to a noticeable decrease of the lifetimes of both series around n⫽46 and to a shift of the 46s1 1 state. Carrying out the excitation via the B( v ⫽3,N⫽0) intermediate level enables a strong enhancement of the intensities of these series and the observation of the nd1 1 series beyond n⫽110 as illustrated in Fig. 5. Unfortunately, the intensity of this series drops by a factor of 10 beyond the 19d3 1 Rydberg state 共i.e., beyond n⫽80). Because of the strong intensity and the regularity of the nd1 1 series between n⫽50 and 70 and the very long lifetimes of the nd1 1 Rydberg states, these states represent the most suitable gerade Rydberg states for subsequent studies by millimeter wave spectroscopy. The widths of the nd1 1 resonances are broader than the experimental resolution. A deconvolution procedure15 indicates that each line consists of

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FIG. 6. Spectra of the np Rydberg series of ortho H2 in the vicinity of the ⫹ ⫹ X 2⌺ ⫹ g ( v ⫽0,N ⫽1) ionization threshold. The spectra were recorded by monitoring the delayed pulsed field ionization following single-photon excitation from the ground state. The traces correspond in ascending order to spectra obtained by applying the field ionization pulse less than 50 ns, 150 ns, 1 ␮s, and 5 ␮s after photoexcitation. The different predissociation behavior of the three N⫽0 – 2 components of the np1 N complex around n ⫽40 is shown in the enlarged view of the spectra presented on the left-hand side. The zero point of the wave number scale corresponds to the position of ⫹ ⫹ the center of gravity of the hyperfine components of the X 2 ⌺ ⫹ g ( v ⫽0,N ⫽1) ground ionic state. FIG. 5. High-resolution spectrum of the ns and nd Rydberg series of ortho ⫹ ⫹ H2 in the vicinity of the X 2 ⌺ ⫹ g ( v ⫽0,N ⫽1) ionization threshold measured with intracavity etalons in the dye lasers. The spectra were recorded in a (1⫹1 ⬘ ) two-photon excitation sequence using the P(1) line of the 1 ⫹ X 1⌺ ⫹ g ( v ⫽0)-B ⌺ u ( v ⫽3) transition in the first excitation step. The positive trace corresponds to the H⫹ 2 ions produced by pulsed field ionization or autoionization and the negative trace to the predissociation signal corresponding to ionization from the excited n⫽2 H fragment. The assignments in the figure are given in the notation nᐉN ⫹ N . The main series corresponds to the nd1 1 series. The zero point of the wave number scale corresponds to the position of the X 1 ⌺ ⫹ g ( v ⫽0,J⫽0) ground neutral state.

two groups of hyperfine components separated by ⬇1.3 GHz, an interval that corresponds closely to the main hyperfine splitting of the ( v ⫹ ⫽0,N ⫹ ⫽1) ionic level. This observation indicates that the core electron is primarily coupled to the H⫹ 2 nuclear spin, which, in turn, implies that the d states have mixed singlet and triplet character. The nd1 1 Rydberg states offer a means to excite the triplet p Rydberg states by further excitation with the millimeter waves. Figure 6 shows pulsed field ionization spectra of n p singlet Rydberg states of H2 below the N ⫹ ⫽1 ionization threshold recorded at a resolution of 0.1 cm⫺1 following single-photon VUV excitation from the ground state. The spectra reveal three series corresponding to the three values of N (N⫽0 – 2) that can be accessed from the N ⵮ ⫽1 level of the ground neutral state. The assignment of the series can be made easily on the basis of their known quantum defects 共see Sec. III A兲 and is indicated above the experimental spectra. Comparison of spectra recorded at delay times of ⭐50 ns, 150 ns, 1 ␮s, and 5 ␮s between laser and electric field pulse 共see Fig. 6兲 leads to the conclusions that 共1兲 the n p1 1 Rydberg states have lifetimes of more than 5 ␮s in the

range n⫽40– 70 where the series could be resolved, 共2兲 the lifetimes of the n p1 0 Rydberg states gradually increase from a few hundreds of nanoseconds to more than 1 ␮s between n⫽40 and 60, and 共3兲 the n p1 2 Rydberg states predissociate more than one order of magnitude faster than the np1 0 states of similar n values. The n p Rydberg states of H2 are predis41 Rydberg states of ⌺ ⫹ sociated by the B ⬘ 1 ⌺ ⫹ u state: u character are subject to a homogeneous 共vibronic兲 predissociation whereas those of ⌸ ⫹ u symmetry undergo a heterogeneous (N dependent兲 predissociation. The different propensities for predissociation of the three N⫽0, 1, and 2 components of the n p1 Rydberg complexes can be explained qualitatively by considering their symmetries. As explained in Sec. III A, the N⫽1 components possess pure ⌸ ⫺ u character and are thus not predissociated. The N⫽0 components, on the other hand, ⫹ ⫹ possess ⌺ ⫹ u and the N⫽2 component both ⌺ u and ⌸ u character and are thus predissociated by the B ⬘ state. The reduced lifetimes of the N⫽2 Rydberg states compared to the N⫽0 components have their origin in the heterogeneous (N dependent兲 predissociation 共see also Sec. III A兲. Because of their nonpredissociative behavior, the n p1 1 states between n⫽40 and 60 represent the most suitable ungerade Rydberg levels for subsequent millimeter wave experiments. The following conclusions can be drawn from the laser experiments summarized above: 共i兲 The only truly long-lived ( ␶ ⭓5 ␮ s) Rydberg states accessible from the ground neutral state using the excitation sequences 共1兲–共3兲 belong to the n p1 1 and nd1 1 series beyond n⫽50. These states are ideal initial states for subsequent excitation with millimeter waves. 共ii兲 The n p1 0 Rydberg states, and possibly also the nd1 2 and nd1 3 states, have lifetimes sufficiently long for millime-

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J. Chem. Phys., Vol. 121, No. 23, 15 December 2004

FIG. 7. Survey millimeter wave spectra of transitions between initial nd1 1 Rydberg states and final np and n f Rydberg states belonging to series con⫹ ⫹ ⫹ verging on the X 2 ⌺ ⫹ g ( v ⫽0,N ⫽1) ground state of ortho H2 . 共a兲 53d →57p/ f , 共b兲 53d→56p/ f , 共c兲 52d→56p/ f , 共d兲 52d→55p/ f , 共e兲 51d →55p/ f , and 共f兲 51d→54p/ f .

ter wave spectra to be recorded at sub-MHz resolution, particularly beyond n⫽55. 共iii兲 The lifetimes of members of the np1 2 series are too short for these states to be observed in millimeter wave spectra either as initial or as final states. 共iv兲 The lifetimes of the p and d Rydberg states of H2 can be strongly reduced by local perturbations caused by low-n interloper Rydberg states. The np1 1 and nd1 1 Rydberg states of H2 between n⫽50 and 70 are sufficiently distant from low-n interlopers so that the intensities and lifetimes vary smoothly with n. 共v兲 The nd1 1 Rydberg states, which have mixed singlet and triplet character, give access to the study of both singlet and triplet np Rydberg states by millimeter wave spectroscopy. B. Millimeter wave spectra and their assignment

Figures 7 and 8 show surveys of the millimeter wave spectra recorded via excitation sequences 共2兲 and 共3兲, respectively. The spectra have been ordered so as to highlight the procedure used to extract the combination differences outlined in Sec. II.C. The main patterns in these spectra are assigned in each case above the top spectrum and conform to the general structure of the p and f Rydberg manifolds described in Sec. III. They also reflect the clustering of the

Rydberg states of molecular hydrogen

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FIG. 8. Survey millimeter wave spectra of transitions between initial np1 1 Rydberg states and final ns and nd Rydberg states belonging to series con⫹ ⫹ ⫹ verging on the X 2 ⌺ ⫹ g ( v ⫽0,N ⫽1) ground state of ortho H2 . 共a兲 57p →62s/d, 共b兲 56p→61s/d, 共c兲 55p→60s/d, 共d兲 55p→59s/d, 共e兲 54p →59s/d, and 共f兲 54p→58s/d.

nd1 N Rydberg states in two groups of components separated by ⬇1300 MHz and corresponding to G ⫹ values of 1/2 and 3/2 共see preceding section兲. Millimeter wave transitions from these two groups of nd1 1 states lead to the excitation of n p1 0 – 2 Rydberg states of both singlet (S⫽0) and triplet (S⫽1) character 共see Fig. 7兲. However, the short lifetimes of the n p1 2 Rydberg states prevent their observation, as explained above. Of the six n f 1 2 – 4 (G ⫹ ⫽1/2, 3/2) series, only the two n f 1 2 (G ⫹ ⫽1/2, 3/2) series can be efficiently excited from the nd1 1 Rydberg states because of the ⌬N⫽⌬ᐉ propensity rule that governs transitions between nonpenetrating Rydberg states 共see Sec. III c and Fig. 2兲. One therefore expects six pairs of line clusters at each n value in the millimeter wave spectra recorded from the nd1 1 Rydberg state, corresponding to transitions from each set of the two nd1 1 (G ⫹ ⫽1/2 and 3/2) components to the n p1 0 (S⫽0), np1 0 (S ⫽1), n p1 1 (S⫽0), n p1 1 (S⫽1), n f 1 2 (G ⫹ ⫽1/2), and n f 1 2 (G ⫹ ⫽3/2) Rydberg states. Five of these six pairs are represented in the spectra of Fig. 7. The sixth doublet, corresponding to transitions to the n p1 0 (S⫽1) states, lies more than 7 GHz below the transitions to the n p1 0 (S⫽0) components and was only recorded at n⫽63. The assignment of ᐉ was confirmed experimentally in a

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FIG. 9. Assignment of the final state orbital angular momentum ᐉ in transitions from an initial nd1 1 to final np/ f Rydberg states from a measurement of the Stark effect induced by an electric field of 40 mV/cm. 共a兲 51d1 1 (G ⫹ ⫽3/2)→55p1 0 (S⫽0), 共b兲 51d1 1 (G ⫹ ⫽3/2)→55p1 1 (S⫽1), and 51d1 1 (G ⫹ ⫽3/2) 共c兲 51d1 1 (G ⫹ ⫽1/2)→55f 1 2 (G ⫹ ⫽1/2), ⫹ ⫹ →55f 1 2 (G ⫽3/2), and 共d兲 51d1 1 (G ⫽3/2)→55p1 1 (S⫽0). The transitions to the f states are strongly shifted and broadenend by the electric field whereas the transitions to the p states remain sharp and are hardly shifted 共see text for additional details兲.

measurement of line shifts and line broadenings induced by an electric field of 40 mV/cm and relied on the different behavior of the p and f states in weak electric fields illustrated by Fig. 9. The transitions to the f states are strongly shifted and broadened by the electric field because of their large polarizability 关Fig. 9共c兲兴. The p states, in contrast, have a much smaller polarizability and are neither strongly shifted nor broadened by weak electric fields. However, the electric field relaxes the ⌬F⫽0,⫾1 selection rule and leads to the observation of additional lines 关Figs. 9共a兲, 9共b兲, and 9共d兲兴. The overall structure of the millimeter wave spectra recorded from the np1 1 (S⫽0) Rydberg states can be explained by similar considerations: Transitions to the nd1 1 (G ⫹ ⫽1/2,3/2) and nd1 2 (G ⫹ ⫽1/2,3/2) states form two pairs of line clusters separated by about 1300 MHz located below an isolated feature corresponding to the transition to the ns1 1 (S⫽0) states 共see Fig. 8兲. In contrast to excitation sequences 共1兲 and 共2兲, sequence 共3兲 does not proceed through levels of mixed singlet and triplet character and does not allow the observation of ns1 1 (S⫽1) states. The assignment of the total angular momentum quantum number F of the initial and final states of the millimeter wave transitions necessitated additional considerations and the careful examination of the structure of each cluster of transitions. Enlarged views of the ten clusters of transitions represented in Fig. 7共c兲 (51d→55p/ f ) are depicted in Fig. 10. Analysis of the combination differences between all lines in these clusters revealed that both groups of nd1 1 levels consist of three hyperfine components. The F quantum number of these six nd hyperfine levels could be assigned on the basis of the patterns observed in the millimeter wave spectra and the ⌬F⫽0,⫾1 (0↔ ” 0) selection rule. The assignment was facilitated by the observation of a perfect correspondence of the spectral patterns observed from each of the two

Osterwalder et al.

FIG. 10. High-resolution millimeter wave spectra of transitions from the hyperfine structure components of the 51d1 1 Rydberg state to the hyperfine structure components of the 55p/ f Rydberg states. The spectra were recorded using measurement times of 3 ␮s. 共a兲 51d1 1 (G ⫹ ⫽3/2)→55p1 0 (S ⫽0), 共b兲 51d1 1 (G ⫹ ⫽1/2)→55p1 0 (S⫽0), 共c兲 51d1 1 (G ⫹ ⫽3/2) →55p1 1 (S⫽1), 共d兲 51d1 1 (G ⫹ ⫽1/2)→55p1 1 (S⫽1), 共D兲 51d1 1 (G ⫹ ⫽3/2)→55f 1 2 (G ⫹ ⫽1/2), 共E兲 51d1 1 (G ⫹ ⫽1/2)→55f 1 2 (G ⫹ ⫽1/2), 共e兲 51d1 1 (G ⫹ ⫽3/2)→55f 1 2 (G ⫹ ⫽3/2), 共f兲 51d1 1 (G ⫹ ⫽1/2)→55f 1 2 (G ⫹ ⫽3/2), 共g兲 51d1 1 (G ⫹ ⫽3/2)→55p1 1 (S⫽0), 共h兲 51d1 1 (G ⫹ ⫽1/2) →55p1 1 (S⫽0). Because the 55p1 0 (S⫽0) Rydberg state is not split by the hyperfine interaction, the 共inverted兲 hyperfine structure of the two G ⫹ components of the initial 51d1 1 state is directly observable in traces 共a兲 and 共b兲.

triplets of nd1 1 hyperfine components: Whenever three, two, or one transitions were observed from the upper triplet of nd1 1 states to a given final state, three, two, or one transitions were also observed from the lower triplet of nd1 1 states to the same final state. For example, the two triplets in panels 共a兲 and 共b兲 of Fig. 10 correspond to transitions from the three F⫽0 – 2 components of the 51d1 1 of each G ⫹ manifold to an F⫽1 final state. This observation implies that each of the two triplets of nd levels is composed of one F ⫽0, one F⫽1, and one F⫽2 state. Moreover, whenever three transitions from either of the two sets of initial nd1 1 levels with F⫽0,1, and 2 are observed to a given final state, this state must be an F⫽1 state. Similarly, the observation of two groups of two transitions to a given final state implies that the final state can be assigned to an F⫽2 state and the initial nd1 1 states to F⫽1 and 2 states, which, in turn, implies that the missing component of each triplet of nd1 1 corresponds to an F⫽0 state. Finally, the observation of only one component of each triplet leads to the conclusion that the final state is either an F⫽0 state 共in which case the initial state is an F⫽1 state兲 or an F⫽3 state 共in which case the initial state is an F⫽2 state兲. A distinction between these two possibilities can be made by analyzing the patterns of transitions to the n p1 1 (S⫽0) final states for which no F⫽3 component exists. Once the assignment of the total angular momentum quantum number F of the six initial nd1 1 levels

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is established, the determination of the F values of the final states of the millimeter wave transitions is straightforward. A similar procedure was employed to assign the F quantum number of the initial and final states of the millimeter wave transitions recorded from the np1 1 (S⫽0) levels. A total of more than 1000 millimeter wave transitions could be unambiguously assigned, representing more than 90% of all observed transitions. From the transition frequencies, the positions, relative to a common origin, of nine s levels between n⫽59 and 62, seventy p levels between n ⫽54 and 64, hundred and five d levels between n⫽51 and 62, and seventy-three f levels between n⫽54 and 64 could be determined with an accuracy of better than 1 MHz. The positions and assignments of the p and f levels, which are at the object of the present study, are listed with their experimental uncertainties in Table VI. Three experimental assignments had to be questionned in the light of the calculations, and the MQDT fit of the p and f Rydberg states included 140 of the 143 levels.

V. THE HYPERFINE STRUCTURE IN THE GROUND STATE OF H2¿

The effective Hamiltonian describing the hyperfine 2 ⫹ structure of the rotational levels of H⫹ 2 in its X ⌺ g ground electronic state can be expressed as H⫽H vib⫹H rot⫹H HFS ,

共5兲

with H HFS⫽aI"L⫹ ⫹b 1F I¿"S¿⫹ 31 c 共 3I z S z⫹ ⫺I"S⫹ 兲 ⫹dS¿"N¿ ⫹e 关 S¿"S¿⫺ 共 S z⫹ 兲 2 兴 ⫹ f 关 I"I⫺ 共 I z 兲 2 兴 ⫽aI¿"L¿⫹bI"S¿⫹cI z S z⫹ ⫹dS¿"N¿⫹e 关 S¿"S¿ ⫺ 共 S z⫹ 兲 2 兴 ⫹ f 关 I"I⫺ 共 I z 兲 2 兴 ,

共6兲

and b⫽b 1F ⫺ (c/3). The validity of this effective Hamiltonian to describe the hyperfine structure of the v ⫹ ⫹ ⫽4⫺8,N ⫹ ⫽1 levels of the X 2 ⌺ ⫹ g state of H2 has been tested by comparison with the transition frequencies between the hyperfine structure levels measured by Jefferts with an accuracy of 1 kHz.47 Moreover, the hyperfine structure constants b, c, and d calculated ab initio by Babb and Dalgarno46 were found to be in excellent agreement with constants derived from the hyperfine structure of nonpenetrating n⫽27 Rydberg states.13 The effective Hamiltonian in Eq. 共6兲 can be expressed in matrix form using several angular momentum coupling basis sets. If the nuclear spin is disregarded, the rotational energy level structure is adequately described by Hund’s coupling case 共b兲,33 in which H rot in Eq. 共5兲 is diagonal with elements B v N ⫹ (N ⫹ ⫹1). Additional consideration of the nuclear spins leads to three coupling cases derived from Hund’s case 共b兲, cases (b␤ J ⫹ ), (b␤ S ⫹ ), and (b␤ N ⫹ ) 共see Introduction for the nomenclature兲. Cases (b␤ J ⫹ ) and (b␤ S ⫹ ), for instance, imply that the rotational levels can be classified in Hund’s case 共b兲 and that the total nuclear spin I is coupled to the total angular momentum without spins J⫹ and the electron spin S⫹ , respectively. Case (b␤ N ⫹ ), in which the nuclear 42,43,45,46

11825

spin is coupled to the core rotation, only becomes a physically plausible limiting case in highly excited rotational levels and is not considered further here. The eigenvalues and eigenfunctions of the hyperfine structure components of the N ⫹ ⫽1 level of H⫹ 2 can be derived by calculating the elements of H HFS in either case (b␤ J ⫹ ) or (b␤ S ⫹ ) and solving the eigenvalue problem. Although possible procedures have been discussed in the literature 共see, e.g., Refs. 23 and 43兲 the derivation of correct expressions for all relevant matrix elements requires some care. The procedure followed in the present study was motivated by the choice of expressing the MQDT frame transformation using the coupling case (e关 b␤ S ⫹ 兴 ) to describe the long-range dissociation channels 关see Eq. 共4兲 in Sec. III兴. However, instead of deriving the matrix elements by evaluating them in the (b␤ S ⫹ ) basis, it turned out to be easier to first derive them in the coupling case (b␤ J ⫹ ) following the procedure established by Mizushima43 and then to transform the matrix into the (b␤ S ⫹ ) basis by applying the (b␤ J ⫹ )↔(b␤ S ⫹ ) frame transformation. The aI"L¿ term in Eq. 共6兲 vanishes in a 2 ⌺ ⫹ g state. The bI"S¿ term has both diagonal and off-diagonal elements in case (b␤ J ⫹ ). The diagonal elements and the off-diagonal elements connecting levels with J ⫹ and J ⫹ ⫾1 are given by Eqs. 共14a兲 and 共14b兲 of Ref. 43, respectively. The cI z S z term also possesses both diagonal and off-diagonal elements in case (b␤ J ⫹ ) and expressions for these elements are listed in Eqs. 共13a兲–共13c兲 of Ref. 43. Off-diagonal elements between levels with (N ⫹ ,J ⫹ ) and (N ⫹ ⫾2,J ⫹ ⫾2) have not been included because they are smaller than the energy separation between the N ⫹ ⫽1 and N ⫹ ⫽3 rotational levels by more than four orders of magnitude and have a negligible effect on the eigenvalues and eigenfunctions. Moreover, off-diagonal elements of bI"S⫹ and cI z S z coupling states with (N ⫹ ,I) and (N ⫹ ⫾1,I⫾1), i.e., elements causing ortho-para mixing, have been neglected in the present analysis, because they only become important in highly excited vibrational levels.48 The term dS⫹ •N⫹ is diagonal in (b␤ J ⫹ ), and the diagonal elements were calculated using Eq. 共7兲:

具 ⌳ ⫹ N ⫹ S ⫹ J ⫹ 兩 S⫹ •N⫹ 兩 ⌳ ⫹ N ⫹ S ⫹ J ⫹ 典 ⫽

关 J ⫹ 共 J ⫹ ⫹1 兲 ⫺S ⫹ 共 S ⫹ ⫹1 兲 ⫺N ⫹ 共 N ⫹ ⫹1 兲兴 . 2

共7兲

The last two terms of Eq. 共6兲 do not make a significant contribution and have been neglected in previous investigations. They have also been neglected here. With these approximations, H HFS is diagonal in N ⫹ and the eigenvalues for the ⫹ v ⫹ ⫽0,N ⫹ ⫽1 level of the X 2 ⌺ ⫹ g ground state of H2 can be expressed as 共see Refs. 43 and 45 and Appendix B兲

冋冉 册

b c d E ⫾ 共 F ⫹ ⫽1/2兲 ⫽⫺ ⫺ ⫺ ⫾ 4 4 4 ⫹

2 共 c⫹2b 兲 2 36

3d c 7b ⫹ ⫺ 4 12 12



2

1/2

,

共8兲

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J. Chem. Phys., Vol. 121, No. 23, 15 December 2004

b c d E ⫾ 共 F ⫹ ⫽3/2兲 ⫽⫺ ⫹ ⫺ ⫾ 4 20 4 5 ⫹ 共 c⫹2b 兲 2 36



冋冉

3d 7 b ⫺ ⫺ 4 60 12



2

with

1/2

,

b c d E 共 F ⫹ ⫽5/2兲 ⫽ ⫹ ⫹ . 2 10 2

共9兲 共10兲

⫹ In the special case of a 2 ⌺ ⫹ g state (⌳ ⫽0), the transformation matrix between the (b␤ S ⫹ ) basis and the (b␤ J ⫹ ) basis was determined to be 共see Appendix B兲

具 ⌳ ⫹ ⫽0N ⫹ S ⫹ G ⫹ 兩 ⌳ ⫹ ⫽0N ⫹ S ⫹ J ⫹ 典 ⫽ 冑共 2J ⫹ ⫹1 兲共 2G ⫹ ⫹1 兲共 ⫺1 兲 ⫺N ⫻



J⫹

N⫹

S⫹

G⫹

I

F⫹



⫹ ⫹S ⫹ ⫹2G ⫹ ⫹I⫺F ⫹

共11兲

.

Using the hyperfine coupling constants reported by Babb and Dalgarno46 for the ( v ⫹ ⫽0,N ⫹ ⫽1) level of H⫹ 2 , the following eigenvalues and eigenfunctions 关expressed as linear combinations of case (b␤ S ⫹ ) basis functions 兩 G ⫹ ,F ⫹ 典 ] can be derived as 共see Appendix B兲 E ⫺ 共 F ⫹ ⫽3/2兲 ⫽⫺930.405 MHz,

共12兲

E ⫺ 共 F ⫹ ⫽1/2兲 ⫽⫺910.804 MHz,

共13兲

E ⫹ 共 F ⫹ ⫽1/2兲 ⫽385.271 MHz,

共14兲

E ⫹ 共 F ⫹ ⫽3/2兲 ⫽481.961 MHz,

共15兲

E 共 F ⫹ ⫽5/2兲 ⫽474.140 MHz,

共16兲





⌿ 共 F ⫽3/2兲 ⫽⫺0.999 878兩 1/2,3/2典 ⫹0.015 608 5 兩 3/2,3/2典 ,

共17兲

⌿ ⫺ 共 F ⫹ ⫽1/2兲 ⫽⫺0.999 246兩 1/2,1/2典 ⫹0.038 824 1 兩 3/2,1/2典 , ⫹

共18兲



⌿ 共 F ⫽1/2兲 ⫽0.038 824 1 兩 1/2,1/2典 ⫹0.999 246兩 3/2,1/2典 ,

共19兲

⌿ ⫹ 共 F ⫹ ⫽3/2兲 ⫽0.015 608 5 兩 1/2,3/2典 ⫹0.999 878 兩 3/2,3/2典 , ⫹

⌿ 共 F ⫽5/2兲 ⫽ 兩 3/2,5/2典 .

共20兲 共21兲

Equations 共17兲–共21兲 contain the information needed to determine the transformation 具 e关 b␤ S ⫹ 兴 兩 eexact典 required by Eq. 共4兲 in Sec. III.D.

␯ i共 E 兲 ⫽



RyM E⫹ i ⫺E



1/2

共23兲

.

Here, E is the total energy, ␯ is the effective principal quantum number, and A is Ham’s scaling function given in Eq. 共2.31兲 of Ref. 49. The spin-rovibronic Rydberg channel i attaches to the core hyperfine level v ⫹ N ⫹ G ⫹ F ⫹ of energy RyM ⫽109 707.42 cm⫺1 is the mass-corrected RydE⫹ i . berg constant for H2 . As stated above, the total angular momentum F and the total parity (⫺1) p with p⫽0 or 1 are the only constants of motion in field-free space. Each channel index i stands for the core state plus the associated electron spin-orbital partial wave j as discussed in Sec. III. For a given set of F and p values, the spin-rovibronic quantum defect matrices of Eq. 共22兲 are constructed from the bodyframe ␩ quantum defects by means of the spin-rovibronic frame transformation tan ␲ ␩ i,i ⬘ ⫽ (F,p)

冕␹

(N ⫹ ) v⫹ 共 R 兲



S⌳⍀

关 具 i (sre) 兩 ␣ (sre) 典

⫻tan ␲ ␩ ᐉ,ᐉ ⬘ 共 E,R 兲 具 ␣ ⬘ (sre) 兩 i ⬘ (sre) 典 兴 (S⌳⍀)

(N ⫹ )

⫻ ␹ v ⫹ ⬘ ⬘ 共 R 兲 dR,

where the dependence of the ion core vibrational wave func(N ⫹ )

tions ␹ v ⫹ (R) on G ⫹ and F ⫹ has been neglected 共but the corresponding hyperfine structure of the ion thresholds E ⫹ i has of course been taken into account兲. The frame transformation elements 具 i (sre) 兩 ␣ (sre) 典 have been defined in Secs. III (S⌳⍀) and V. ␩ ᐉ,ᐉ ⬘ (E,R) are the electronic body-frame quantum defect matrix elements. We use the ␩ quantum defects here rather than the more familiar ␮ quantum defects to avoid nonphysical states with n⭐ᐉ that could occur in series converging to rovibrationally excited H⫹ 2 and their interactions with the high n states of interest here. The ␩ quantum defects are taken to depend only on S, ⌳, and ⍀ but not on I nor on J. The electronic quantum defect matrix contains elements that couple states of different ᐉ values, i.e., we do not assume the orbital angular momentum to be conserved in the body frame. The frame transformation in Eq. 共24兲 can be expressed in a more compact form tan ␲ ␩ i,i ⬘ ⫽ (F,p)



␣ ( v sre)

关 具 i ( v sre) 兩 ␣ ( v sre) 典 tan ␲ ␩ ᐉ,ᐉ ⬘ 共 E 兲

⫻具 ␣ ⬘ ( v sre) 兩 i ⬘ ( v sre) 典 兴 , VI. QUANTUM DEFECT THEORY OF HYPERFINE STATES A. Spin-rovibronic quantum defect matrices

The quantization condition for bound levels in MQDT is used here in its customary form37



det



tan ␲␯ i 共 E 兲 ␦ ii ⬘ (F,p) ⫹tan ␲ ␩ i,i ⬘ ⫽0 A 共 E,ᐉ i 兲

共22兲

共24兲

( v sre)

共25兲

which emphasizes the fact that the integration over the vibrational coordinate itself has the character of a frame transformation. i ( v sre) and ␣ ( v sre) symbolize the spin-rovibronic asymptotic and short-range channels, respectively. In the following subsection, we discuss the body-frame quantum defects that are relevant for the present problem, and in particular their dependences on R and E, and on the clamped-nuclei quantum numbers S, ⌳, and ⍀.

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B. General discussion of ungerade symmetry body-frame quantum defects in H2

The separation of subsequent members n and n⫹1 of a Rydberg series near n⫽60 amounts to 1.0 cm⫺1 . Thus, for a quantitative prediction of the observed series at the level of the present experimental accuracy of better than 1 MHz, the quantum defects must be known to better than 3⫻10⫺5 . Conversely, quantum defects could be determined to within this accuracy from the experiment. The (n * ) ⫺3 scaling law for energy intervals between Rydberg states converts this accuracy into energies of 0.8 cm⫺1 for n⫽2 共lowest p states兲 or of 0.1 cm⫺1 for n⫽4 共lowest f states兲. Therefore, any contribution to the quantum defects which is known to yield energy shifts one or more orders of magnitude smaller than these energies in the lowest Rydberg states need not be included in the quantitative analysis of the n⬇60 Rydberg states. For example, the electron spin fine structure in the ⫺1 2 2p 3 ⌸ ⫺ u state (c state兲 is of the order of 0.12 cm . The ⍀ splittings of the ␩ pp triplet quantum defects are therefore expected to make contributions to the level energies of the n⬇60p states of the order of 0.000 004 cm⫺1 or 130 kHz, i.e., smaller than the resolution of the present measurements, and they will therefore be neglected below. On the other hand, the singlet-triplet splittings in the 4 f levels have been resolved by Fourier transform infrared 共FTIR兲 and infrared laser difference frequency spectroscopy14,34 and were found to be of the order of 0.06 cm⫺1 . The scaling law predicts that their contributions to the n⬇60f states must be of the order of 0.000 02 cm⫺1 or 500 kHz, which implies that the difference between the ␩ f f quantum defects for S⫽0 and 1 should be observable experimentally. We shall show in Sec. VII that this is indeed the case. The body-frame quantum defects relevant in the present context correspond to diagonal elements ␩ S⌳⍀ with ⌳⫽0 and 1 and S⫽0 and 1 共singlet and triplet pp , components兲, respectively, and to diagonal elements ␩ S⌳⍀ ff ⌳⫽0 – 3, again with S⫽0 and 1. Off-diagonal quantum de, ⌳⫽0,1 and S⫽0,1, which account for fect elements ␩ S⌳⍀ pf the p- f mixing arising from the nonspherical nature of the ion core, must also be included. As already stated, all these body-frame quantum defect matrix elements depend in principle on the quantum numbers S, ⌳, and ⍀ and are functions of the internuclear distance R and of the energy E. The R dependences are known to lead to vibronic coupling involving channels v ⫹ and v ⫹ ⬘ 关Eq. 共24兲兴. The energy dependences determine the exact positions of low-n, high-v ⫹ interloper levels with respect to the highn, low-v ⫹ series, and therefore have a strong influence on the occurrence of local vibronic perturbations mediated by the R dependence of the quantum defects. We now discuss the various symmetries in turn. Singlet p ␴ and p ␲ . The singlet p ␴ and p ␲ Rydberg series dominate the one-photon absorption spectrum of H2 and have been studied extensively by Takezawa50 and Herzberg and Jungen33 many years ago in the region n * ⬇15– 30. Jungen and Atabek51 published a quantum defect analysis of the n * ⫽2 – 4 members of the same series. In the most recent theoretical determination of the p␴ and p␲ quantum defects, Jungen and Ross52 used the best available

FIG. 11. Body-frame ␩ (R) pp quantum defects for p channels in H2 as functions of the internuclear distance 共in bohr兲. Full lines: initial values from quantum chemical potential curve calculations 共see text for additional details兲. Circles: fitted values, see Sec. VII. The vertical bars indicate the classical turning points of the vibrational motion for v ⫹ ⫽0 共connected by arrows兲 and for v ⫹ ⫽5, respectively.

quantum-chemical potential energy curves for n * ⫽2 – 4 to extract the combined R- and E-dependences of the quantum defects. Here, we follow their method of representing the quantum defects by three numerical functions ␩ (R), 关 ⳵ ␩ / ⳵ E 兴 (R), and 关 ⳵ 2 ␩ / ⳵ E 2 兴 (R) which are defined point by point on an adequate grid of R values. These functions 共after transformation from the ␮ to the ␩ defect representation兲 are illustrated in Figs. 11–13 共full lines兲. These same functions, when used in a spinless rovibronic multichannel quantum defect treatment with a vibrational basis v ⫹ ⫽0 – 10 关Eqs. 共22兲–共25兲 without spins兴, reproduce 116 singlet p levels observed in Ref. 33 between

FIG. 12. 关 ⳵ ␩ pp / ⳵ E 兴 (R) in Ry⫺1 . Full lines: initial values from quantum chemical potential curve calculations. Circles: local expansion analogous to Eq. 共31兲, except for 1 ⌺ ⫹ u where the adjusted values are given 共see text for additional details兲. The vertical bars indicate the classical turning points of the vibrational motion for v ⫹ ⫽0 共connected by arrows兲 and for v ⫹ ⫽5, respectively.

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FIG. 13. 关 ⳵ 2 ␩ pp / ⳵ E 2 兴 (R) in Ry⫺2 . Full lines: initial values from quantum chemical potential curve calculations. Circles: local expansion analogous to Eq. 共31兲 共see text for additional details兲. The vertical bars indicate the classical turning points of the vibrational motion for v ⫹ ⫽0 共connected by arrows兲 and for v ⫹ ⫽5, respectively.

123 058 and 124 451 cm⫺1 for N c/d ⫽0 c , 1 c , 1 d , 2 c , and 2 d 共where c/d represents Kronig’s parity index兲 with a mean deviation of 1.1 cm⫺1 . The largest deviations amount to several cm⫺1 and affect the low-n interlopers with v ⫹ ⬎0. In order to optimize our initial quantum defect functions, a least-squares fitting procedure based on the observed levels of Refs. 50 共low-n/high-v ⫹ levels兲 and33 共high-n/low-v ⫹ levels兲 was carried out. In this fit, a minimal vibrational basis ( v ⫹ ⫽0 – 6) was used in view of the subsequent timeconsuming hyperfine calculations. We found that a slight adjustment of the derivative functions 关 ⳵ ␩ / ⳵ E 兴 (R) for ⌳⫽0 and 1 near R⫽2a 0 was necessary to reduce the mean deviation to 0.7 cm⫺1 , close to the experimental accuracy of the measurements of Ref. 33, and to bring the calculated vibrational interlopers into their correct positions. Figure 14

FIG. 14. Predicted ungerade vibronic level structure of singlet and triplet p states ( v ⫹ ⫽1 – 6) in the vicinity of the v ⫹ ⫽0 ionization threshold of H2 . Rotational levels N c/d (c/d Kronig’s symmetry兲 belonging to the same vibronic level nᐉ␭, v ⫹ are connected by oblique lines. Levels of mixed character are not labeled. The gray area marks the region investigated by millimeter wave spectroscopy.

Osterwalder et al.

represents an energy level diagram which displays the v ⫹ ⬎1 vibrational interloper structure ordered according to N c/d . Levels 0 c , 1 d , 2 c , . . . have positive total parity and may interact with the v ⫹ ⫽0 levels of interest here. The 4 p ␴ , v ⫹ ⫽4,N⫽2 c level situated at 124 451.9 cm⫺1 is located within 2 cm⫺1 of the region of interest here and is likely 共and will be found below兲 to perturb the levels studied in this work. The modified 关 ⳵ ␩ / ⳵ E 兴 (R) function for 1 ⌺ ⫹ u symmetry resulting from the fit is represented in Fig. 12 by circles. Triplet p ␴ and p ␲ . The triplet p series in H2 have not been observed beyond n * ⫽4. We have used the ab initio 3 ⫹ ⌺ u potential energy curves calculated by Detmer, Schmelcher, and Cederbaum53 to derive 3 ⌺ ⫹ u quantum defects in a similar fashion as described in Ref. 52 for the singlet components. The 3 ⌸ u quantum defect functions have previously been described54 and are used at this point without adjustment. Strictly speaking, and as already mentioned, there are three slightly different quantum defect functions ␩ (S⫽1,⌳⫽1,⍀) with ⍀⫽0, 1, and 2, respectively. This fine pp structure is of the order of 5⫻10⫺6 on the quantum defect scale and may be neglected here. The triplet ␩ pp quantum defect functions and their energy dependences are displayed in Figs. 11–13. They have been used to predict the v ⫹ ⭓1 interloper structure represented in Fig. 14. No triplet interloper is predicted to come close to the levels studied in this work, and the triplet p functions shown in Figs. 11–13 should therefore be accurate enough for the initial calculations of the v ⫹ ⫽0 hyperfine structure. Singlet and triplet f ␭ (␭⫽0 – 3). The f electron in H2 is nearly nonpenetrating. The f ␭ (␭⫽0 – 3) quantum defects are known through the work of Ref. 34 共see also Ref. 14兲 where it was shown that the 4 f manifold of rovibronic levels up to v ⫹ ⫽3 can be predicted from first principles to within about 0.5 cm⫺1 in the framework of a long-range multipole and dispersion force model for the electron-core interaction. This type of model was first applied to nonpenetrating molecular Rydberg states in Ref. 55 and refined and extended in Ref. 7 where electron-core interaction terms up to 1/r 8 were included in a systematic fashion. Expressions for the required ab initio R-dependent polarizability and hyperpolarizability tensors and for the quadrupole, hexadecapole, and hexacontadecapole moments of the ion core can be found in the literature.56 –59 They may be used to set up an electroncore long-range and dispersion force interaction potential V(r, ␪ ,R) where r and ␪ are the electron radial coordinate and the polar angle defined with respect to the fixed molecular frame. Specifically, the f quantum defects have been determined from the radius- and angle-dependent interaction potential V(r, ␪ ,R) in the following way: Neglecting ᐉ mixing, V(r, ␪ ,R) was first converted into 2 ᐉ 共 ᐉ⫹1 兲 (␭) ⫹ 具 ᐉ␭ 兩 V 共 r, ␪ ,R 兲 兩 ᐉ␭ 典 V ᐉᐉ 共 r,R 兲 ⫽⫺ ⫹ r r2

共26兲

for each R and ␭⬅⌳ value through integration over the angular electron coordinates ␪ and ␾. Second, Rydberg binding energies ⑀ nᐉ␭ (R) were evaluated for each R value by direct numerical integration of the Schro¨dinger equation for the

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radial motion of the electron in the potential of Eq. 共26兲. Third, ␩ quantum defects were determined for each R value by use of the clamped-nuclei version of the MQDT expression 关Eq. 共22兲兴 for bound levels, tan ␲ 关 ␯ nᐉ␭ 共 R 兲兴 ⫹tan ␲ 关 ␩ nᐉ␭ 共 R 兲兴 ⫽0, A 关 ⑀ nᐉ␭ 共 R 兲 ,ᐉ 兴

共27兲

where

␯ nᐉ␭ 共 R 兲 ⫽ 关 ⑀ nᐉ␭ 共 R 兲兴 ⫺1/2

共28兲

represents the effective principal quantum number and A is Ham’s scaling function as before. ␯ nᐉ␭ (R) is calculated from the energies ⑀ nᐉ␭ (R) evaluated in the preceding step 关note that the energies in Eq. 共28兲 are in Rydbergs and must not be mass corrected51兴. Equation 共27兲 can then be used to obtain the quantum defect for any desired n value. Using the resulting values ␩ nᐉ␭ (R) for n⫽4, 10, and 55, ᐉ⫽3, and ⌳⫽␭ (R), 关 ⳵ ␩ (S⌳⍀) / ⳵ E 兴 (R), and ⫽0 – 3, the desired ␩ (S⌳⍀) ff ff 2 (S⌳⍀) 2 / ⳵ E 兴 (R) functions can finally be extracted for 关⳵ ␩ f f each R value. The long-range multipole and dispersion force model does not account for exchange interactions nor for relativistic effects, and consequently it predicts quantum defects that are the same irrespective of the S and ⍀ quantum numbers. The functions ␩ (⌳) f f (R) and their energy dependence thus obtained are represented in Figs. 16 –18. We must also consider the possibility that the v ⫹ ⫽0 high-n f levels observed in our work may be perturbed by vibronic interaction with vibrational interlopers as discussed above for the p manifolds. Indeed, a ⌬ v ⫹ ⫽1 perturbation involving 5 f ( v ⫹ ⫽0) and 4 f ( v ⫹ ⫽1) and caused by near coincidences of levels has been observed in the infrared 6g →5 f emission spectrum of H2 in Ref. 60, and causes level shifts of the order of 1 cm⫺1 . Using appropriate n * scaling, perturbations of the n⬇60f levels by up to 0.008 cm⫺1 or 250 MHz can be predicted. Based on the quantum defect functions from Figs. 16 –18, the interloper structure including vibrational channels up to v ⫹ ⫽4 was calculated. The result is displayed in Fig. 15. Two levels with N⫽3 d , namely, 4 f ( v ⫹ ⫽3) with N ⫹ ⫽5 and 7 f ( v ⫹ ⫽1) with N ⫹ ⫽1, are indeed predicted to lie within a few cm⫺1 of the energy region studied here. Their effect will be investigated in the calculations including nuclear spins 共see Sec. VII兲. However, in view of previous experience with f states, and of their overall small quantum defects, we are confident that the positions of these possible perturbers is predicted sufficiently well by the theoretical quantum defects and energy derivatives of Figs. 16 –18, so that no adjustment of the derivatives need be made.

FIG. 15. Predicted ungerade vibronic level structure of f states ( v ⫹ ⫽1 – 4) in the vicinity of the v ⫹ ⫽0 ionization threshold of H2 . The levels shown belong to 6 f and 7 f , v ⫹ ⫽1, 5 f , v ⫹ ⫽2 and 4 f , v ⫹ ⫽3. They are not labeled invidually for the sake of clarity. The gray area marks the region investigated by millimeter wave spectroscopy.

polarization and multipolar field components have dropped to vanishingly small values in comparison with the dominating term. Simultaneously, the energy differences of the various vibration–spin-rotation core levels should also be vanishingly small compared to the Coulomb term, although they become dominant at very large distances. Although no particular radius r 0 appears in expressions such as Eqs. 共24兲 and 共25兲, its existence is actually implied. An important aim of this work is to use multichannel quantum defect theory with unprecedented requirements of accuracy. Before turning to the discussion of the actual calculations, it therefore appears necessary to establish the reliability of this method. Although qualitative rules exist for estimating under what circumstances the frame transformation method can be

C. Validity of the frame transformation method

The concept of frame transformations plays a central role in the present work and in molecular quantum defect theory in general, but is known to represent an approximation as discussed, for example, in Chaps. 7.5 and 9 of Ref. 61. The frame transformation approach hinges on the requirement that there should be a region outside the molecular core, say near an electron distance r⫽r 0 , where the core field has become purely Coulombic, and in particular the

FIG. 16. Body-frame ␩ f f (R) quantum defects for f channels in H2 as functions of the internuclear distance 共in bohr兲. Full lines: initial values from the long-range force model. Circles: fitted values. The vertical bars indicate the classical turning points of the vibrational motion for v ⫹ ⫽0 共connected by arrows兲 and for v ⫹ ⫽5, respectively.

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⫹ E⫹ i ␦ i,i ⬘ ⫹V i,i ⬘ 共 r 兲 ⫽E i ␦ i,i ⬘ ⫹

兺␣ 具 i 兩 ␣ 典 (ᐉ) V ᐉᐉ(␭)共 r,R 兲 具 ␣ 兩 i ⬘ 典 (ᐉ) ,

共29兲

E⫹ i

where represent the energies of the vibrational rotational hyperfine levels of the ion core. With the electron kinetic energy operator added, Eq. 共29兲 defines a system of coupled differential equations which does not assume the electron-ion scattering at short range to be diagonal in ␣, or, in other words, to be diagonal in I, S, ⌳, and ⍀, because the frame transformation is applied directly to the potential V for each r value rather than to the reaction matrix K. For comparison, the MQDT secular equation restricted to a single ᐉ value and expressed in the short-hand notation of Eq. 共29兲 reads



det FIG. 17. 关 ⳵ ␩ f f / ⳵ E 兴 (R) in Ry⫺1 calculated from the long-range force model. The vertical bars indicate the classical turning points of the vibrational motion for v ⫹ ⫽0 共connected by arrows兲 and for v ⫹ ⫽5, respectively.

used 共see, e.g., Ref. 61兲, there is to our knowledge no quantitative general criterion available for assessing the validity of the frame transformation approximation. A criterion, formulated in Ref. 62, exists but is limited to the specific case of rotational-electronic coupling. The f electron in H2 is very nearly nonpenetrating and can be described quite accurately by its motion in the Coulomb field with added polarization and multipole terms 关Eq. 共26兲兴. The local character of the potential of Eq. 共26兲 can therefore be exploited to set up a system of coupled differential equations. Denoting the asymptotic laboratory frame channels by i ( v sre) and the body-frame channels by ␣ ( v sre) as before, we convert Eq. 共26兲 to the laboratory frame. Adding the asymptotic target energies E ⫹ i and dropping the superscripts v sre from here on, leaving just the relevant index ᐉ, we obtain

tan ␲␯ i 共 E 兲 ␦ i,i ⬘ ⫹ A 共 E,ᐉ i 兲



兺␣ 具 i 兩 ␣ 典 (ᐉ) tan ␲ ␩ ᐉᐉ(␭)共 R 兲 具 ␣ 兩 i ⬘ 典 (ᐉ)

⫽0.

共30兲

We have made a series of test calculations for the 55f levels with F⫽0, 1, 2, and 3 which allow a comparison between the results of the frame transformation MQDT calculations with those obtained by solving the coupled equations. To this end, a coupled equations code was developed based on Johnson’s renormalized Numerov method.63 In our calculations, we used a grid of 20 000 points extending out to 10 000a 0 with a step size varying between 0.125 and 0.7a0 . Only v ⫹ ⫽0 target channels were included in these test calculations. Table V summarizes the results thus obtained. The data indicate that the differences between the two sets of calculations are of the order of 0.000 01 cm⫺1 or 0.3 MHz at most, which means that the frame transformation approximation can be used safely in the present context. Note that the MQDT calculations are four to five orders of magnitude faster than the calculations based on the coupled equations. Without this gain in calculational speed, the least-squares fits that will be described in Sec. VII would have been prohibitively time consuming. Note also that for the p states which are penetrating, no local potential of the type of Eq. 共26兲 can be defined, and the option of solving coupled equations to replace the approximate MQDT calculations is not available. VII. RESULTS AND DISCUSSION

Table VI lists the positions of all assigned levels with their experimental uncertainties „see Sec. II; rather than experimental uncertainties ␦ E/(hc cm⫺1 ) themselves, the where W table lists the values 10⫺5 ⫻W 1/2 ⫺1 2 ⫽1/关 ␦ E/(hc cm ) 兴 represents the weights used in the least-squares fits… and the relevant quantum numbers n, ᐉ, N ⫹ , N and either G ⫹ and F for the f states or S and F for the p states. For the least-squares fitting procedure, each of (S⌳⍀) (R) has been represented by a the diagonal functions ␩ ᐉᐉ second-order polynomial (S⌳⍀) (S⌳⍀) ␩ ᐉᐉ 共 R 兲 ⫽ ␩ ᐉᐉ 共 R⫽R 0 兲 ⫹

FIG. 18. 关 ⳵ 2 ␩ f f / ⳵ E 2 兴 (R) in Ry⫺2 calculated from the long-range force model. The vertical bars indicate the classical turning points of the vibrational motion for v ⫹ ⫽0 共connected by arrows兲 and for v ⫹ ⫽5, respectively.



(S⌳⍀) ⳵ ␩ ᐉᐉ 共 R⫺R 0 兲 ⳵R

(S⌳⍀) 1 ⳵ 2 ␩ ᐉᐉ 共 R⫺R 0 兲 2 ⫹¯ 2 ⳵R2

共31兲

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TABLE V. Comparison of level positions 共in cm⫺1 ) determined by MQDT and coupled equations calculations. The positions are given relative to that ⫺1 of the X 1 ⌺ ⫹ have g ( v ⫽0,J⫽0) ground neutral state, whereby 124 439 cm been subtracted from the actual values. The calculations were carried out using the following theoretical values: ionization energy IE/(hc) ⫽124 417.512 cm⫺1 共Ref. 64兲, 关 E ⫹ ( v ⫹ ⫽0,N ⫹ ⫽1)⫺E ⫹ ( v ⫹ ⫽0,N ⫹ ⫽0) 兴 /(hc)⫽58.2337 cm⫺1 共Ref. 65兲 and the hyperfine structure of the v ⫹ ⫽0, N ⫹ ⫽1 level from Ref. 46. MQDT

CC

CC-MQDT

N⫽2,G ⫹ ⫽3/2

F⫽0: 共6 channels兲 0.479 281 0.479 283

0.000 002

N⫽2,G ⫹ ⫽1/2 N⫽2,G ⫹ ⫽3/2 N⫽2,G ⫹ ⫽3/2 N⫽3,G ⫹ ⫽3/2

F⫽1: 共18 channels兲 0.432 430 0.432 433 0.479 281 0.479 283 0.479 332 0.479 335 0.500 044 0.500 035

0.000 003 0.000 002 0.000 002 ⫺0.000 009

N⫽2,G ⫹ ⫽1/2 N⫽2,G ⫹ ⫽1/2 N⫽3,G ⫹ ⫽1/2 N⫽2,G ⫹ ⫽3/2 N⫽2,G ⫹ ⫽3/2 N⫽3,G ⫹ ⫽3/2 N⫽4,G ⫹ ⫽3/2 N⫽3,G ⫹ ⫽3/2

F⫽2: 共29 channels兲 0.432 430 0.432 433 0.432 788 0.432 791 0.453 280 0.453 271 0.478 982 0.478 984 0.479 332 0.479 335 0.499 195 0.499 186 0.482 928 0.482 916 0.500 044 0.500 035

0.000 003 0.000 003 ⫺0.000 009 0.000 002 0.000 003 ⫺0.000 009 ⫺0.000 012 ⫺0.000 009

N⫽2,G ⫹ ⫽1/2 N⫽4,G ⫹ ⫽1/2 N⫽3,G ⫹ ⫽1/2 N⫽3,G ⫹ ⫽1/2 N⫽2,G ⫹ ⫽3/2 N⫽2,G ⫹ ⫽3/2 N⫽4,G ⫹ ⫽3/2 N⫽4,G ⫹ ⫽3/2 N⫽3,G ⫹ ⫽3/2 N⫽3,G ⫹ ⫽3/2

F⫽3: 共37 channels兲 0.432 788 0.432 791 0.438 328 0.438 315 0.453 154 0.453 144 0.453 280 0.453 271 0.478 074 0.478 077 0.478 982 0.478 984 0.482 928 0.482 916 0.484 197 0.484 185 0.498 834 0.498 824 0.499 195 0.499 186

0.000 003 ⫺0.000 013 ⫺0.000 010 ⫺0.000 009 0.000 003 0.000 002 ⫺0.000 012 ⫺0.000 012 ⫺0.000 010 ⫺0.000 009

spanning nine grid points from 1.2a 0 to 2.8a 0 with R 0 ⫽2a 0 . This region largely covers the range of R values relevant here which corresponds primarily to v ⫹ ⫽0 共see Figs. 11–13兲. The grid values falling outside this range have been left unchanged during the fitting procedure. We found it convenient to represent the ⌳ structure of the ᐉ⫽3 quantum defects of Eq. 共31兲 as

␩ (S⫽0,⌳⍀) 共 R 0 兲 ⫽a 0 ⫹a 1 ⌳⫹a 2 ⌳ 2 ⫹a 3 ⌳ 3 , ff

共32兲

the four a i constants enabling the independent variation of the four ⌳ components. The ⍀ dependence of the ␩ f f quantum defects was neglected. Similarly the small triplet-singlet splitting for ᐉ⫽3 was represented as

␩ (S⫽1,⌳⍀) 共 R 0 兲 ⫺ ␩ (S⫽0,⌳⍀) 共 R 0 兲 ⫽b 0 ⫹b 1 ⌳⫹b 2 ⌳ 2 ⫹b 3 ⌳ 3 . ff ff

共33兲

The p- f ᐉ-mixing interaction represented by the off-diagonal (R) (⌳⫽0 and 1兲 was exquantum defect element ␩ (S⌳⍀) pf pressed as

␩ (S⌳⍀) 共 R 0 兲 ⫽c 0 ⫹c 1 ⌳, pf

共34兲

neglecting its dependence on the electron spin S and on ⍀. Expressions analogous to Eq. 共32兲 were also used for ⳵ ␩ / ⳵ R

11831

and ⳵ 2 ␩ / ⳵ R 2 and the energy-derivative functions 关 ⳵ ␩ / ⳵ E 兴 (R) and 关 ⳵ 2 ␩ / ⳵ E 2 兴 (R). The parameters corresponding to the energy derivatives were kept fixed at their ab initio values 共see Figs. 11–13 and 16 –18兲 in the leastsquares fitting procedure. The initial hyperfine structure calculations were carried out with the quantum defects determined in Sec. VI B. The singlet-triplet splittings in the f manifold and the p- f interaction were set to zero, and vibrational bases v ⫹ ⫽0 – 6 and v ⫹ ⫽0 – 1 were used for ᐉ⫽1 and for ᐉ⫽3, respectively. The total number of channels in the spin-rovibronic MQDT calculations thus amounted to 47 for F⫽0, 127 for F⫽1, 177 for F⫽2, and 200 for F⫽3. The initial calculations were carried out separately for the p and f manifolds of states. These calculations immediately led to unambiguous assignments of the F values of all observed levels which fully confirmed the experimental assignments discussed in Sec. IV. Note that the assignments were made entirely independently in Zurich and in Orsay, thus giving even stronger support to the correctness of the results. Once the assignments were established, the energy of the reference experimental level 关 51d1 1 (G ⫹ ⫽1/2),F⫽0, see Sec. II C兴 with respect to the center of gravity of the hyperfine structure of the v ⫹ ⫽0, N ⫹ ⫽1 level was adjusted slightly, leaving the hyperfine splittings unchanged at this stage, such as to optimize the agreement between observed and calculated levels. In this way, sixty-nine p levels were calculated with a mean deviation observed-calcculated of 0.0014 cm⫺1 共42 MHz兲, and 71 f levels were reproduced with a mean deviation of 0.0004 cm⫺1 共13 MHz兲. In a next stage, least-squares fitting calculations were carried out including both p and f channels in which a total of 12 parameters were adjusted, namely, all four ␩ pp quantum defects at R 0 ⫽2a 0 , three ␩ f f quantum defects 关 a 0 (R 0 ) and a 2 (R 0 ) and the singlet-triplet splitting constant b 0 (R 0 )], one ␩ p f ᐉ-mixing quantum defect 关 c 0 (R 0 ) 兴 , three H⫹ 2 hyperfine splitting constants (b, c, and d), and the relative position of the v ⫹ ⫽0, N ⫹ ⫽1, G ⫹ ⫽1/2, F ⫹ ⫽1/2 ionic level. The weights W given to the observed levels were W ⫽1/关 ␦ E/(hc cm⫺1 ) 兴 2 where ␦ E/(hc cm⫺1 ) represents the experimental error 共see Table VI兲. The fitting procedure proved a little delicate in the beginning, particularly because of the perturbing 4p ␴ , v ⫹ ⫽4, N⫽2 c singlet level. For example, an adjustment of the ␩ (S⫽0,⌳⫽0,⍀⫽0) quantum defect by only ⫹0.001 shifts the pp 60p ␴ , v ⫹ ⫽0 levels down by 0.001 cm⫺1 and the 4p ␴ , v ⫹ ⫽4 level down by 3.9 cm⫺1 共120 GHz兲, placing it incorrectly right into the midst of the levels studied here 共see Fig. 14兲. Conversely, a corresponding shift of ␩ (S⫽0,⌳⫽0,⍀⫽0) in pp the opposite direction displaces the 4p ␴ , v ⫹ ⫽4, N⫽2 c singlet level too far from the levels observed here, thus underestimating its effect on the high-n, v ⫹ ⫽0 p levels and resulting in a systematic trend of the residuals observedcalculated of the order of 0.000 03 cm⫺1 or 0.9 MHz over the range of n levels studied in the present work. For this reason, the least-squares fits of the hyperfine structure had to be carried out in alternation with the determination of energy

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J. Chem. Phys., Vol. 121, No. 23, 15 December 2004

TABLE VI. Observed and calculated energy levels (cm⫺1 ). Approximate descriptiona 54p1 0 54p1 1 54p1 1 54p1 1 54p1 1 54p1 1 55p1 0 55p1 1 55p1 1 55p1 1 55p1 1 55p1 1 56p1 0 56p1 1 56p1 1 56p1 1 56p1 1 56p1 1 56p1 1 57p1 0 57p1 1 57p1 1 57p1 1 57p1 1 57p1 1 57p1 1 58p1 0 58p1 1 58p1 1 58p1 1 58p1 1 58p1 1 58p1 1 59p1 0 59p1 1 59p1 1 59p1 1 59p1 1 59p1 1 59p1 1 60p1 0 60p1 1 60p1 1 60p1 1 60p1 1 60p1 1 60p1 1 61p1 0 61p1 1 61p1 1 61p1 1 61p1 1 61p1 1 62p1 0 62p1 1 62p1 1 62p1 1 62p1 1 62p1 1 63p1 0 63p1 1 63p1 1 63p1 1 64p1 0 64p1 1 64p1 1 64p1 1

共0兲 共1兲 共1兲 共0兲 共0兲 共0兲 共0兲 共1兲 共1兲 共0兲 共0兲 共0兲 共0兲 共1兲 共1兲 共1兲 共0兲 共0兲 共0兲 共0兲 共1兲 共1兲 共1兲 共0兲 共0兲 共0兲 共0兲 共1兲 共1兲 共1兲 共0兲 共0兲 共0兲 共0兲 共1兲 共1兲 共1兲 共0兲 共0兲 共0兲 共0兲 共1兲 共1兲 共1兲 共0兲 共0兲 共0兲 共0兲 共1兲 共1兲 共0兲 共0兲 共0兲 共0兲 共1兲 共1兲 共0兲 共0兲 共0兲 共1兲 共0兲 共0兲 共0兲 共0兲 共1兲 共0兲 共0兲

F

Observed energyb

10⫺5 冑W c

Observedcalculated

Approximate descriptiona

F

1 1 2 1 0 2 1 2 0 1 0 2 1 1 2 0 1 0 2 1 1 2 0 1 0 2 1 1 2 0 1 0 2 1 1 2 0 1 0 2 1 1 2 0 1 0 2 1 1 2 1 0 2 1 1 2 1 0 2 1 1 0 2 1 2 1 0

4.405 40 4.581 07 4.581 68 4.791 80 4.791 91 4.791 96 5.775 66 5.941 43 5.941 62 6.141 52 6.141 63 6.141 69 7.072 94 7.228 31 7.228 91 7.229 13 7.419 69 7.419 80 7.419 87 8.302 34 8.448 56 8.449 16 8.449 33 8.631 29 8.631 41 8.631 48 9.468 49 9.606 18 9.606 76 9.606 95 9.780 85 9.780 97 9.781 05 10.575 67 10.705 37 10.705 95 10.706 14 10.872 54 10.872 67 10.872 75 11.627 80 11.750 01 11.750 58 11.750 78 11.910 19 11.910 32 11.910 41 12.628 45 12.743 63 12.744 21 12.897 31 12.897 46 12.897 53 13.580 85 13.689 50 13.690 10 13.837 12 13.837 28 13.837 36 14.234 01 14.732 60 14.732 76 14.732 85 15.353 34 15.450 45 15.586 46 15.586 63

3.00 3.00 1.80 3.00 2.00 1.90 0.71 0.70 0.75 0.70 0.68 0.70 0.70 0.70 0.68 0.68 0.70 0.67 0.70 0.68 0.65 0.63 0.60 0.67 0.67 0.65 0.77 0.61 0.50 0.55 0.75 0.60 0.73 0.46 0.46 0.45 0.46 0.46 0.42 0.45 0.86 0.42 0.43 0.43 0.43 0.43 0.43 0.43 0.43 0.43 0.43 0.43 0.43 0.38 0.20 0.20 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38

⫺0.000 001 ⫺0.000 002 ⫺0.000 003 ⫺0.000 001 0.000 000 0.000 001 ⫺0.000 001 0.000 002 0.000 016 0.000 000 ⫺0.000 004 0.000 003 0.000 000 0.000 000 0.000 002 0.000 043 ⫺0.000 002 ⫺0.000 013 0.000 002 0.000 012 ⫺0.000 001 0.000 007 ⫺0.000 005 0.000 003 ⫺0.000 003 0.000 007 0.000 007 0.000 004 ⫺0.000 002 0.000 002 ⫺0.000 001 ⫺0.000 012 0.000 004 0.000 004 0.000 001 0.000 000 0.000 001 ⫺0.000 006 ⫺0.000 014 ⫺0.000 001 0.000 012 0.000 008 ⫺0.000 002 0.000 007 ⫺0.000 004 ⫺0.000 018 0.000 001 0.000 010 0.000 005 0.000 003 0.000 006 0.000 006 0.000 000 ⫺0.000 075 0.000 010 0.000 008 0.000 013 0.000 016 0.000 016 0.000 009 0.000 022 0.000 019 0.000 023 0.000 012 ⫺0.000 006 ⫺0.000 003 ⫺0.000 002

64p1 1 共0兲

2

1 54f 1 2 ( 2 ) 1 54f 1 2 ( 2 ) 1 54f 1 4 ( 2 ) 3 54f 1 2 ( 2 ) 3 54f 1 2 ( 2 ) 3 54f 1 2 ( 2 ) 3 54f 1 2 ( 2 ) 3 54f 1 2 ( 2 ) 1 55f 1 2 ( 2 ) 1 55f 1 2 ( 2 ) 1 55f 1 4 ( 2 ) 3 55f 1 2 ( 2 ) 3 55f 1 2 ( 2 ) 3 55f 1 2 ( 2 ) 3 55f 1 2 ( 2 ) 3 55f 1 2 ( 2 ) 3 55f 1 4 ( 2 ) 1 56f 1 2 ( 2 ) 1 56f 1 2 ( 2 ) 1 56f 1 4 ( 2 ) 3 56f 1 2 ( 2 ) 3 56f 1 2 ( 2 ) 3 56f 1 2 ( 2 ) 3 56f 1 2 ( 2 ) 3 56f 1 4 ( 2 ) 1 57f 1 2 ( 2 ) 1 57f 1 2 ( 2 ) 3 57f 1 2 ( 2 ) 3 57f 1 2 ( 2 ) 3 57f 1 2 ( 2 ) 3 57f 1 2 ( 2 ) 3 57f 1 4 ( 2 ) 1 58f 1 2 ( 2 ) 1 58f 1 2 ( 2 ) 3 58f 1 2 ( 2 ) 3 58f 1 2 ( 2 ) 3 58f 1 2 ( 2 ) 1 59f 1 2 ( 2 ) 1 59f 1 2 ( 2 ) 3 59f 1 2 ( 2 ) 3 59f 1 2 ( 2 ) 3 59f 1 2 ( 2 ) 3 59f 1 2 ( 2 ) 1 60f 1 2 ( 2 ) 1 60f 1 2 ( 2 ) 3 60f 1 2 ( 2 ) 3 60f 1 2 ( 2 ) 3 60f 1 2 ( 2 ) 1 61f 1 2 ( 2 ) 1 61f 1 2 ( 2 ) 3 61f 1 2 ( 2 )

Observed energyb

10⫺5 冑W c

Observedcalculated

1

15.586 73 4.631 22

0.38 2.00

0.000 006 ⫺0.000 001

2

4.631 58

1.20

0.000 002

3

4.637 37

0.86

0.000 013

3

4.676 97

0.46

⫺0.000 003

3

4.677 78

0.60

⫺0.000 013

2

4.677 82

1.50

⫺0.000 009

2

4.678 14

1.20

⫺0.000 003

1

4.678 19

1.50

0.000 018

1

5.987 71

0.67

0.000 003

2

5.988 07

0.67

0.000 005

3

5.993 51

0.67

⫺0.000 002

3

6.033 41

0.67

0.000 007

3

6.034 25

0.60

⫺0.000 010

2

6.034 28

0.67

0.000 002

2

6.034 61

0.67

⫺0.000 005

1

6.034 64

0.67

0.000 013

2

6.038 18

0.67

0.000 039

1

7.272 16

0.67

⫺0.000 005

2

7.272 52

0.67

⫺0.000 005

3

7.277 66

0.67

⫺0.000 009

3

7.317 83

0.60

⫺0.000 008

3

7.318 70

0.67

⫺0.000 005

2

7.319 07

0.67

⫺0.000 003

1

7.319 09

0.67

0.000 009

2

7.322 34

0.60

0.000 033

1

8.489 60

0.60

⫺0.000 004

2

8.489 97

0.60

0.000 004

3

8.535 26

0.55

⫺0.000 001

3

8.536 10

0.32

⫺0.000 031

2

8.536 13

0.46

⫺0.000 012

1

8.536 53

0.50

0.000 009

2

8.539 51

0.55

0.000 029

1

9.644 59

0.75

⫺0.000 004

2

9.644 96

0.67

⫺0.000 004

3

9.690 23

0.50

⫺0.000 019

2

9.691 11

0.43

⫺0.000 020

1

9.691 51

0.60

⫺0.000 017

1

10.741 41

0.43

⫺0.000 047

2

10.741 79

0.40

⫺0.000 001

3

10.786 98

0.43

⫺0.000 018

2

10.787 86

0.43

⫺0.000 017

2

10.788 23

0.43

⫺0.000 059

1

10.788 28

0.43

⫺0.000 013

1

11.783 76

0.43

0.000 003

2

11.784 12

0.43

⫺0.000 054

3

11.829 34

0.43

⫺0.000 019

2

11.830 21

0.43

⫺0.000 025

1

11.830 66

0.43

⫺0.000 012

1

12.775 31

0.38

0.000 008

2

12.775 67

0.38

0.000 035

3

12.820 83

0.38

⫺0.000 036

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J. Chem. Phys., Vol. 121, No. 23, 15 December 2004

Rydberg states of molecular hydrogen

11833

TABLE VI. 共Continued.兲 Approximate descriptiona 3

61f 1 2 ( 2 ) 3 61f 1 2 ( 2 ) 3 61f 1 2 ( 2 ) 1 61f 1 2 ( 2 ) 1 62 f 1 2 ( 2 ) 3 62 f 1 2 ( 2 ) 3 62 f 1 2 ( 2 ) 3 62 f 1 2 ( 2 ) 1 63f 1 2 ( 2 ) 1 63f 1 2 ( 2 ) 3 63f 1 2 ( 2 ) 3 63f 1 2 ( 2 ) 3 63f 1 2 ( 2 ) 1 64f 1 2 ( 2 ) 1 64f 1 2 ( 2 ) 3 64f 1 2 ( 2 ) 3 64f 1 2 ( 2 ) 3 64f 1 2 ( 2 ) 3 64f 1 2 ( 2 ) 3 64f 1 2 ( 2 )

F

Observed energyb

10⫺5 冑W c

Observedcalculated

2

12.821 68

0.38

⫺0.000 058

2

12.822 12

0.38

⫺0.000 073

1

12.822 18

0.38

⫺0.000 019

1

13.719 24

0.38

0.000 008

2

13.719 60

0.38

0.000 024

3

13.764 80

0.38

0.000 019

2

13.765 66

0.38

0.000 014

1

13.766 13

0.38

0.000 056

1

14.618 59

0.38

0.000 027

2

14.618 95

0.38

0.000 041

3

14.664 12

0.38

0.000 026

1

14.665 44

0.38

0.000 034

2

14.667 24

0.38

0.000 026

1

15.476 09

0.38

0.000 028

2

15.476 42

0.38

0.000 011

3

15.521 55

0.38

⫺0.000 024

2

15.522 42

0.38

⫺0.000 002

1

15.522 85

0.38

⫺0.000 056

2

15.522 95

0.38

⫺0.000 008

2

15.524 52

0.38

⫺0.000 045

The notation used is nᐉN N⫹ (S) for p states and nᐉN N⫹ (G ⫹ ) for f states. 1 Energy in cm⫺1 with respect to the 51d1 1 (G ⫹ ⫽ 2 ),F⫽0 level. c The weights used in the fittings are W⫽1/关 ␦ E/(hc cm⫺1 ) 兴 2 . a

b

dependences of the quantum defects described above, until convergence was achieved. The final calculation yielded a rms error of 0.000 020 cm⫺1 or 600 kHz for 140 observed levels, with a standard deviation of 0.96 共see Table VI兲. The standard deviation is sufficiently close to 1 so that one can assert that the present theoretical approach has exploited the complete information content of the data set. It also indicates that the

FIG. 19. Comparison of the observed level positions of the ungerade Rydberg states at n⫽55 below the v ⫹ ⫽0, N ⫹ ⫽1 ground state of ortho H⫹ 2 on the left-hand side with the calculated positions on the right-hand side. The energy scale has been expanded by a factor of 120 compared to Fig. 1.

FIG. 20. Comparison of the observed level positions of the ungerade Rydberg states at n⫽55 below the v ⫹ ⫽0, N ⫹ ⫽1 ground state of ortho H⫹ 2 on the left-hand side with the calculated positions on the right-hand side. The energy scale has been expanded by a factor of 380 compared to Fig. 1. The experimental accuracy of ⬇450 kHz in this range is delimited by the two vertical arrows.

estimation of the experimental errors is realistic.28 The agreement between theory and experiment is illustrated in Figs. 1, 2, 19, and 20 which compare the observed level positions at n⫽55 on the left-hand side with the calculated positions on the right-hand side. The energy scale of Fig. 2 has been expanded by a factor of 10 and those of Figs. 19 and 20 by factors of 120 and 380 compared to Fig. 1, respectively. Figure 21 finally has the same scale as Fig. 2 but shows the level pattern for n⫽63 instead of n⫽55. It can be seen that, because of the interaction with the 4p ␴ , v ⫹ ⫽4, N⫽2 singlet level situated just 4 cm⫺1 above the energy range displayed, the hyperfine levels associated with 63p1 2 (S⫽0) are pushed down into the group of levels corresponding to 63f 1 2 (G ⫹ ⫽1/2) and 63f 1 4 (G ⫹ ⫽1/2). The local perturbation predicted for F⫽3 has not yet been observed. The experimental accuracy of 450 kHz in this range only becomes visible on the scale of Fig. 20 where it is delimited

FIG. 21. Comparison of the observed level positions of the f Rydberg states at n⫽63 below the v ⫹ ⫽0, N ⫹ ⫽1 ground state of ortho H⫹ 2 on the left-hand side with the calculated positions on the right-hand side.

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11834

TABLE VII. Fitted MQDT parameters. Standard deviation 共140 fitted levels兲; 0.96; rms deviation, 0.000 020 cm⫺1 or 600 kHz. Initial value ⫺1

Ionization energy (cm

1

1

42.270 539共10兲

⫹ ⫹ ⫺1 X 2⌺ ⫹ g ( v ⫽0,N ⫽1) ion hyperfine coupling constants (cm ) 0.029 359 3 0.029 364共7兲 b c 0.004 285 7 0.004 288共22兲 d 0.001 415 0 0.001 399共6兲

␩ pp quantum defects (R⫽2.0a 0 ) ⌺⫹ u ⌸u 3 ⫹ ⌺u 3 ⌸u 1 1

0.189 300 ⫺0.078 784 0.461 428 0.059 349

0.189 020共6兲 ⫺0.079 252(7) 0.458 008共7兲 0.061 325共7兲

0.015 166 0.000 128 ⫺0.003 142 0.000 144 0 0 0 0

0.015 360共11兲 0.000 128 ⫺0.003 166(3) 0.000 144 0.000 019共11兲 0 0 0

␩ f f quantum defects (R⫽2.0a 0 ) 关Eq. 关Eq. 关Eq. 关Eq. 关Eq. 关Eq. 关Eq. 关Eq.

共32兲兴 共32兲兴 共32兲兴 共32兲兴 共33兲兴 共33兲兴 共33兲兴 共33兲兴

␩ p f quantum defects (R⫽2.0a 0 ) c 0 关Eq. 共34兲兴 c 1 关Eq. 共34兲兴

0 0

TABLE VIII. Orders of magnitude of various contributions to the calculated level energies. cm⫺1

Fitted value

)

1

T ⬁ ( v ⫹ ⫽0,N ⫹ ⫽1,G ⫹ ⫽ 2 ,F ⫹ ⫽ 2 )⫺51d 1 (G ⫹ ⫽ 2 ),F⫽0

a0 a1 a2 a3 b0 b1 b2 b3

Osterwalder et al.

J. Chem. Phys., Vol. 121, No. 23, 15 December 2004

0.000 622共32兲 0

by the two vertical arrows. In Figs. 19–21, the positions of the energy levels have been referenced to the X 1 ⌺ ⫹ g (v ⫽0,J⫽0) neutral ground state using theoretical values for the ionization energy 共124417.412 cm⫺1 from Ref. 64兲 and for the rotational energy of the ( v ⫹ ⫽0,N ⫹ ⫽1) level of H⫹ 2 (58.2337 cm⫺1 from Ref. 65兲. Table VII compares the initial ab initio parameters with the values obtained in the final fit. The position of the v ⫹ ⫽0, N ⫹ ⫽1, G ⫹ ⫽1/2, F ⫹ ⫽1/2 level with respect to the reference level 51d1 1 (G ⫹ ⫽1/2,F⫽0) is determined with a precision of 10⫺5 cm⫺1 , or 300 kHz. Using the wave number interval of 58.2337 cm⫺1 between the v ⫹ ⫽0, N ⫹ ⫽1 and the v ⫹ ⫽0, N ⫹ ⫽0 levels reported by Moss65 enables one to relate the position of the 51d1 1 (G ⫹ ⫽1/2,F⫽0) level to that of the v ⫹ ⫽0, N ⫹ ⫽0, J ⫹ ⫽1/2 ionic level; a value of 15.9328 cm⫺1 is obtained 关with the 51d1 1 (G ⫹ ⫽1/2),F⫽0 level lying by this amount above the ionization energy兴 that could be used in a future determination of the adiabatic ionization energy of the H2 molecule. The fitted hyperfine coupling constants of the v ⫹ ⫽0, ⫹ N ⫽1 level are compatible with the ab initio values of Babb and Dalgarno,46 but appear to be about ten times less precise than those determined by Fu et al.13 Interestingly, the analysis in Ref. 13 did not take vibrational channel interactions into account and involved a fit of the residual stray fields.66 It is therefore not certain that the accuracy achieved in Ref. 13 is better than ours. The ␩ pp and ␩ f f quantum defects are only changed by at most 4%, the deviation between ab initio and fitted values being largest for the 3 ⌸ u component. The singlet-triplet splitting (⫺b 0 ) of the quantum defects is determined here from

Correction of initial ␩ pp quantum defects Correction of initial ␩ f f quantum defects H⫹ 2 not exactly in case b␤ S ⫹ , effect on frame transformation Vibronic mixing ‘‘⳵ / ⳵ R’’ (⌬ v ⫹ ⫽1) for f levels Vibronic mixing ‘‘⳵ / ⳵ R’’ (⌬ v ⫹ ⫽4) for p levels Singlet-triplet splittings ( f levels兲 p- f interaction Absolute accuracy of the measurement Correction to initial H⫹ 2 hyperfine levels Frame transformation approximation

0.002 0.0002 0.000 15 0.000 15 0.000 03 0.000 02 0.000 02 0.000 02 0.000 01 0.000 01

MHz 60 6 4.5 4.5 0.9 0.6 0.6 0.6 0.3 0.3

the resolved splitting of the two F components associated with several f levels, however, with a fairly large error bar. The value of ⫺b 0 ⫽⫺0.000 019(11) compares favorably with the value of ⫺0.000 021 that can be estimated by multiplying the mean 4 f ⌳ singlet-triplet interval calculated by Meyer67 共viz., ⫹0.073 cm⫺1 at R⫽2a 0 ) by 4 3 /(2Ry). The ␩ p f quantum defect at R⫽2a 0 is also determined here. Its 1 determination rests on the fact that the N⫽2, 1 ⌺ ⫹ u , and ⌸ u levels lie in the middle of the f manifold 共see Figs. 2 and 21兲 and thus perturb the latter. The p- f mixing has a particular significance as it induces rotational p- f ⌬N ⫹ ⫽⫺2 and ⫺4 autoionization and also influences the angular distribution of photoelectrons. The orders of magnitude of various contributions to the improvement of the calculated level energies are listed in Table VIII ordered according to their importance. The largest contribution stems from the adjustment of the ␩ pp and to a lesser extent the ␩ f f quantum defects despite the almost quantitative nature of the ab initio predictions 共see Table VII兲. An unexpected result of this work is that the very small deviation of the ionic levels from (b␤ S ⫹ ) coupling 关see Eqs. 共17兲–共21兲兴 propagates through the frame transformation and leads to level shifts that are as large as those originating from the ⌬ v ⫹ ⫽1 vibronic couplings and more than ten times larger than the experimental precision. The exchange interaction in the f manifold and the p- f interaction lead to level shifts at the limit of detection of the present experiments. A possible breakdown of the frame transformation would lead to effects that are not detectable at present. Finally, the ab initio hyperfine splittings of Babb and Dalgarno46 did not need any significant adjustment in the fit. VIII. CONCLUSIONS

In this work we have developed the theoretical and experimental methodology to study the energy level structure of highly excited molecular Rydberg states at an unprecedented level of detail. Fundamental insights in the channel structures and the coupling hierarchy, including interactions involving the nuclear spins, have been gained as detailed in Sec. VII. The theoretical methods would be generally applicable to other diatomic molecules provided that a sufficient body of ab initio quantum defect functions are available. Experi-

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J. Chem. Phys., Vol. 121, No. 23, 15 December 2004

Rydberg states of molecular hydrogen

11835

TABLE IX. Quantum numbers characterizing the coupling cases and the angular momentum transformations.

mentally, studying other molecules at this degree of resolution will require repeating the procedure, described in detail in Sec. II, of identifying long-lived Rydberg states. Our experience is that sufficiently long-lived Rydberg states exist in all molecules but do not always belong to the low-ᐉ manifolds of Rydberg states.68 In the case of H2 , the method followed has not yet exhausted its potential: in particular, the procedure of building combination differences required by the current lack of a quantitative analysis of the s and d Rydberg states has limited the precision of the data set 共see Sec. II C兲. A significant progress will become possible as soon as the least-squaresfitting procedure can be applied to fit the MQDT parameters directly to the transition frequencies.

Quantum numbers common to all coupling cases

Coupling case

Distinct quantum numbers

b␤ S a␤ J ˜e e关 a␤ J ⫹ 兴 e关 b␤ S ⫹ 兴

N, ⌳, S, G ⌳, S, J, ⍀, q ⌳ ⫹ , S ⫹ , J ⫹ , ⍀ ⫹ , J, q ⫹ , p ⫹ ᐉ, s, j ⌳ ⫹ , S ⫹ , J ⫹ , ⍀ ⫹ , F ⫹ , q ⫹ , p ⫹ , ᐉ, s, j N ⫹ , ⌳ ⫹ , S ⫹ , G ⫹ , F ⫹ , q ⫹ , p ⫹ , ᐉ, s, j

I, F, M F , p

关see Eq. 共4兲兴 are listed below, and Table IX lists the quantum numbers that are specific to the different cases 共second column兲 as well as those that are common to all cases 共third column兲.

ACKNOWLEDGMENTS

This work was supported financially by the ETH Zurich and the Swiss National Science Foundation. A.O. and A.W. thank the Laboratoire Aime´ Cotton for the hospitality during longer periods of research during their Ph.D. and Diploma research, respectively.

共a兲 具 a␤ J 兩˜e典

具 a␤ J 兩˜e典 ⫽ 具 S⌳⍀ 兩 ⍀ ⫹ J ⫹ j 典 ␦ II ⬘ ␦ M I M I ⬘ ␦ FF ⬘ ␦ M F M F ⬘

共A2兲

APPENDIX A: FRAME TRANSFORMATION

with 具 S⌳⍀ 兩 ⍀ ⫹ J ⫹ j 典 given in Eq. 共12兲 of Ref. 38. The coupling case (e ˜) was introduced to facilitate the derivation of the overall transformation. It can be derived from case (e关 a␤ J ⫹ 兴 ) by interchanging the ordering of the coupling of j and I and is, strictly speaking, neither a case 共a兲 nor a case 共e兲.

The explicit expressions for the various contributions to the overall frame-transformation matrix between cases (a␤ J ) and (e关 b␤ S ⫹ 兴 )

具 ␣ sre兩 i sre典 ⫽⌺ 具 a␤ J 兩˜e典具˜e兩 e关 a␤ J ⫹ 兴 典具 e关 a␤ J ⫹ 兴 兩 e关 b␤ S ⫹ 兴 典 ⫻具 e关 b␤ S ⫹ 兴 兩 eexact典

共A1兲

(b) 具˜e兩 e关 a␤ J ⫹ 兴 典

具˜e兩 e关 a␤ J ⫹ 兴 典 ⫽ 冑共 2F ⫹1 兲共 2J⫹1 兲共 ⫺1 兲 ⫹

F ⫹ ⫹I⫹J ⫹ ⫹2 j



F⫹

I

J

j



J ⫹ 关 1⫹ 共 ⫺1 兲 p⫺q 2 F

⫹ ⫹p ⫺q ⫹ ⬘ ⬘



兴 关 1⫹⌬ 1 共 ⫺1 兲 p⫺q ⫺S 共 1⫹⌬ 1 兲

⫹ ⫹J ⫹ ⫹ᐉ



,

共A3兲

with ⌬ 1 ⫽ ␦ ⌳ ⫹ 0 ␦ ⍀ ⫹ 0 . 共c兲 具 e关 a␤ J ⫹ 兴 兩 e关 b␤ S ⫹ 兴 典

具 e关 a␤ J ⫹ 兴 兩 e关 b␤ S ⫹ 兴 典 ⫽ 冑共 2J ⫹ ⫹1 兲共 2N ⫹ ⫹1 兲共 2G ⫹ ⫹1 兲共 ⫺1 兲 ⍀ 关 1⫹ 共 ⫺1 兲 p⫺q ⫻ 2

⫹ ⫹p ⫺q ⫹ ⬘ ⬘

⫹ ⫺2S ⫹ ⫺I⫺F ⫹

兴 关 1⫹ ␦ ⌳ ⫹ 0 共 ⫺1 兲 p⫺q



J⫹

N⫹

S⫹

G⫹

I

F⫹

⫹ ⫹N ⫹ ⫹ᐉ

冑共 1⫹ ␦ ⌳ ⫹ 0 兲共 1⫹⌬ 1 兲



⫹ 兴 N

⌳⫹



S⫹

J⫹

⍀ ⫹ ⫺⌳ ⫹

⫺⍀ ⫹



.

共A4兲

The last transformation 具 e关 b␤ S ⫹ 兴 兩 eexact典 is calculated numerically by diagonalizing the hyperfine Hamiltonian H HFS of the H⫹ 2 ion core as explained in Sec. V and Appendix B. For completeness, we also give here the transformation 具 b␤ S 兩 a␤ J 典 ,

具 b␤ S 兩 a␤ J 典 ⫽ 冑共 2J⫹1 兲共 2N⫹1 兲共 2G⫹1 兲共 ⫺1 兲 ⍀⫺2S⫺I⫺F ⫻

关 1⫹⌬ 4 共 ⫺1 兲 p⫺q

⫹ ⫹N



冑共 1⫹⌬ 2 兲共 1⫹⌬ 3 兲共 1⫹⌬ 4 兲

冋冉



J

N

S

G

I

F

N

S

J



⍀⫺⌳

⫺⍀





关 1⫹ 共 ⫺1 兲 p⫺q 2

⫹⌬ 3 共 ⫺1 兲 p⫺q⫺S⫺J



⫹ ⫹p ⫺q ⫹ ⬘ ⬘



N

S

J



⫺⍀⫺⌳



冊册

,

共A5兲

with ⌬ 2 ⫽ ␦ ⌳ ⫹ 0 ␦ ⌳0 ␦ ⍀0 , ⌬ 3 ⫽(1⫺ ␦ ⌳ ⫹ 0 ) ␦ ⌳0 (1⫺ ␦ ⍀0 ), and ⌬ 4 ⫽ ␦ ⌳ ⫹ 0 ␦ ⌳0 . Downloaded 28 Jan 2005 to 129.132.217.154. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

11836

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J. Chem. Phys., Vol. 121, No. 23, 15 December 2004

APPENDIX B: DIAGONALIZATION OF THE HYPERFINE HAMILTONIAN OF H2¿

When expressed in the case (b␤ J ⫹ ) basis, H HFS takes the following form for the N ⫹ ⫽1 levels 共the order of the basis vectors is 兩 J ⫹ F ⫹ 典 ⫽ 兩 1/2,1/2典 , 兩1/2,3/2典, 兩3/2,1/2典 兩3/2,3/2典, 兩3/2,5/2典兲:



b c ⫺ ⫺d 3 3

0

&b c ⫹ 3 3&

0

b c ⫺ ⫹ ⫺d 6 6

0

&b c ⫹ 3 3&

0

0

冑5b 3

0

冑5b 3

5b c d ⫺ ⫹ 6 6 2





0

冑5c 6

0



0

冑5c

0

6

0

0

0

c d b ⫺ ⫺ ⫹ 3 15 2

0

0

0

c d b ⫹ ⫹ 2 10 2



共B1兲

.

⫹ In the special case of a 2 ⌺ ⫹ g state (⌳ ⫽0), the transformation matrix between the (b␤ S ⫹ ) basis with wave functions

兩 ⌳ ⫹N ⫹S ⫹G ⫹典 ⫽

兺,M

M N⫹





G⫹

N

兺 M ,M S⫹



G

M N⫹

冑共 2F ⫹ ⫹1 兲共 2G ⫹ ⫹1 兲共 ⫺1 兲 ⫺N ⫹ ⫹G ⫹ ⫺M F I



F⫹

M G⫹

⫺M F ⫹

冊冉

S⫹

I

G⫹

M S⫹

MI

⫺M G ⫹



⫹ ⫺S ⫹ ⫹I⫺M G ⫹

兩 IM I 典 兩 S ⫹ M S ⫹ 典 兩 ⌳ ⫹ ⫽0 典 兩 N ⫹ ⌳ ⫹ ⫽0M N ⫹ 典 ,

共B2兲

and the (b␤ J ⫹ ) basis with wave functions

兩 ⌳ ⫹N ⫹S ⫹J ⫹典 ⫽

兺 M ,M J⫹





J

I

兺 M ,M



M J⫹

N⫹

S⫹

冑共 2F ⫹ ⫹1 兲共 2J ⫹ ⫹1 兲共 ⫺1 兲 ⫺J ⫹ ⫹I⫺M F

I

F⫹

MI

⫺M F ⫹

冊冉

N⫹

S⫹

J⫹

M N⫹

M S⫹

⫺M J ⫹



⫹ ⫺N ⫹ ⫹S ⫹ ⫺M J ⫹

兩 IM I 典 兩 S ⫹ M S ⫹ 典 兩 ⌳ ⫹ ⫽0 典 兩 N ⫹ ⌳ ⫹ ⫽0M N ⫹ 典

共B3兲

2 ⫹ ⫹ is given by Eq. 共11兲 in Sec. V. For the N ⫹ ⫽1 rotational levels of the H⫹ 2 ⌺ g (S ⫽1/2,⌳⫽0,I⫽1) state, the transformation matrix takes the form

J⫹ F⫹

1/2 1/2

1/2

1/2



1/2

3/2

0

3/2

1/2

2& 3

3/2

3/2

0

冑5

3/2

5/2

0

0

G



1 3

1/2 3/2

3/2 1/2

3/2 3/2

3/2 5/2

0

2& 3

0

0

冑5

0



2 3

0

3

0

3

1 3

0

0

0

2 3

0

0

0

1

Transforming H HFS in Eq. 共B1兲 using Eq. 共11兲 leads to the following expression in the (b␤ S ⫹ ) basis 共the order of the basis vectors is 兩 G ⫹ F ⫹ 典 ⫽ 兩 1/2,1/2典 , 兩1/2,3/2典, 兩3/2,1/2典 兩3/2,3/2典, 兩3/2,5/2典兲: Downloaded 28 Jan 2005 to 129.132.217.154. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

J. Chem. Phys., Vol. 121, No. 23, 15 December 2004



Rydberg states of molecular hydrogen

&d 3

c d ⫺b⫺ ⫹ 3 3

0

0

c d ⫺b⫺ ⫺ 3 6

0

0

b c 5d ⫺ ⫺ 2 6 6

0

0

0

b 13c d ⫹ ⫺ 2 30 3

0

0

0

c d b ⫹ ⫹ 2 10 2

c 3&

⫹ 0 0

c 3&

&d 3 ⫺

c 6 冑5



0

冑5d 3



0 ⫺

c 6 冑5



This matrix was used to derive Eqs. 共17兲–共21兲 in Sec. V which represent the correct expansions of the wave function ⫹ ⫹ of the hyperfine levels of the X 2 ⌺ ⫹ g ( v ⫽0,N ⫽1) ground ⫹ state of ortho H2 as linear combinations of (b␤ S ⫹ ) basis functions. T. A. Miller and R. S. Freund, Adv. Magn. Reson. 9, 49 共1977兲, and references therein. 2 W. Lichten and T. Wik, J. Chem. Phys. 69, 5428 共1978兲. 3 W. Lichten, T. Wik, and T. A. Miller, J. Chem. Phys. 71, 2441 共1979兲. 4 L. Jo´zefowski, C. Ottinger, and T. Rox, J. Mol. Spectrosc. 163, 381 共1994兲. 5 L. Jo´zefowski, C. Ottinger, and T. Rox, J. Mol. Spectrosc. 163, 398 共1994兲. 6 C. Ottinger, T. Rox, and A. Sharma, J. Mol. Spectrosc. 163, 414 共1994兲. 7 W. G. Sturrus, E. A. Hessels, P. W. Arcuni, and S. R. Lundeen, Phys. Rev. A 38, 135 共1988兲. 8 W. G. Sturrus, E. A. Hessels, P. W. Arcuni, and S. R. Lundeen, Phys. Rev. Lett. 61, 2320 共1988兲. 9 P. W. Arcuni, Z. W. Fu, and S. R. Lundeen, Phys. Rev. A 42, 6950 共1990兲. 10 E. A. Hessels, F. J. Deck, P. W. Arcuni, and S. R. Lundeen, Phys. Rev. A 41, 3663 共1990兲. 11 P. W. Arcuni, E. A. Hessels, and S. R. Lundeen, Phys. Rev. A 41, 3648 共1990兲. 12 W. G. Sturrus, E. A. Hessels, P. W. Arcuni, and S. R. Lundeen, Phys. Rev. A 44, 3032 共1991兲. 13 Z. W. Fu, E. A. Hessels, and S. R. Lundeen, Phys. Rev. A 46, R5313 共1992兲. 14 D. Uy, C. M. Gabrys, T. Oka, B. J. Cotterell, R. J. Stickland, C. Jungen, and A. Wu¨est, J. Chem. Phys. 113, 10143 共2000兲. 15 A. Osterwalder, R. Seiler, and F. Merkt, J. Chem. Phys. 113, 7939 共2000兲. 16 F. Merkt and A. Osterwalder, Int. Rev. Phys. Chem. 21, 385 共2002兲. 17 A. Osterwalder, F. Merkt, and C. Jungen 共unpublished results兲. 18 J. M. Gilligan and E. E. Eyler, Phys. Rev. A 46, 3676 共1992兲. 19 The nuclear spins of the two H atoms are assumed to be coupled to form a resultant total nuclear spin I, i.e., we take I to be a good quantum number and neglect ortho-para mixing which is known to be a good approximation in low vibrational levels 共see Ref. 48 and references therein兲. 20 G. Herzberg, Molecular Spectroscopy and Molecular Structure, Vol. I Spectra of Diatomic Molecules 共Krieger, Malabar, 1991兲. 21 H. Lefebvre-Brion and R. W. Field, The Spectra and Dynamics of Diatomic Molecules 共Elsevier, Amsterdam, 2004兲. 22 C. H. Townes and A. L. Schawlow, Microwave Spectroscopy 共Dover Publications, Dover, 1975兲. 23 R. A. Frosch and H. M. Foley, Phys. Rev. 88, 1337 共1952兲. 24 H. J. Wo¨rner, U. Hollenstein, and F. Merkt, Phys. Rev. A 68, 032510 共2003兲. 25 S. Gerstenkorn, P. Luc, and J. Verges, Atlas du Spectre d’Absorption de la mole´cule de l’Iode 共Editions du CNRS, Orsay, 1993兲. 1

0

冑5d 3

0



.

11837

共B4兲

26

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J. Chem. Phys., Vol. 121, No. 23, 15 December 2004

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