Benchmarking Theory with an Improved Measurement of the Ionization and Dissociation Energies of H_{2}.

The dissociation energy of H_{2} represents a benchmark quantity to test the accuracy of first-principles calculations. We present a new measurement of the energy interval between the EF ^{1}Σ_{g}^{+}(v=0,N=1) state and the 54p1_{1} Rydberg state of H_{2}. When combined with previously determined intervals, this new measurement leads to an improved value of the dissociation energy D_{0}^{N=1} of ortho-H_{2} that has, for the first time, reached a level of uncertainty that is 3 times smaller than the contribution of about 1 MHz resulting from the finite size of the proton. The new result of 35 999.582 834(11) cm^{-1} is in remarkable agreement with the theoretical result of 35 999.582 820(26) cm^{-1} obtained in calculations including high-order relativistic and quantum-electrodynamics corrections, as reported in the following Letter [M. Puchalski, J. Komasa, P. Czachorowski, and K. Pachucki, Phys. Rev. Lett. 122, 103003 (2019)PRLTAO0031-900710.1103/PhysRevLett.122.103003]. This agreement resolves a recent discrepancy between experiment and theory that had hindered a possible use of the dissociation energy of H_{2} in the context of the current controversy on the charge radius of the proton.

Heavy Rydberg states: large amplitude vibrations.

Extremely large vibrational amplitude (≈8700 a.u.) heavy Rydberg levels in the HH[combining macron]1Σ+g state, located only 25 cm-1 below the ion-pair dissociation limit, are reported. The calculations are done using a hybrid log derivative/multichannel quantum defect approach that accounts for predissociation and is capable of dealing with any number of long-range closed channels, and of providing positions and widths for the heavy Rydberg resonances. In this case, resonance positions can be reproduced qualitatively using a simple diabatic model (however, the resonance widths cannot). Absolute quantum defects are derived for the vibrational series ranging from ν = 0 to ν = 2010. The influence of the Coulomb potential and continuity of heavy Rydberg behavior throughout the 1Σ+g manifold of states is demonstrated.

The fundamental rotational interval of para-H2 (+) by MQDT-assisted Rydberg spectroscopy of H2.

Transitions from selected nd Rydberg states of H2 to n'p/f Rydberg series converging on the lowest two (N(+) = 0 and 2) rotational levels of the X(+) (2)Σg (+) (v(+) = 0) ground state of para-H2 (+) have been measured in the range 1-7.4 THz using a laser-based, pulsed, narrow-band source of submillimeter-wave radiation. The analysis of the spectra by multichannel quantum-defect theory (MQDT) has allowed a complete interpretation of the fine structures of the Rydberg series and their dependence on the principal quantum number. The extrapolation of the series to their limits with MQDT has enabled the determination of the first rotational interval of para-H2 (+), which is 174.236 71(7) cm(-1) (5 223 485.1(2.3) MHz).

Determination of the binding energies of the np Rydberg states of H2, HD, and D2 from high-resolution spectroscopic data by multichannel quantum-defect theory.

Multichannel quantum-defect theory (MQDT) is used to calculate the electron binding energies of np Rydberg states of H2, HD, and D2 around n = 60 at an accuracy of better than 0.5 MHz. The theory includes the effects of rovibronic channel interactions and the hyperfine structure, and has been extended to the calculation of the asymmetric hyperfine structure of Rydberg states of a heteronuclear diatomic molecule (HD). Starting values for the eigenquantum-defect parameters of MQDT were extracted from ab initio potential-energy functions for the low-lying p Rydberg states of molecular hydrogen and subsequently refined in a global weighted fit to available experimental data on the singlet and triplet Rydberg states of H2 and D2. The electron binding energies of high-np Rydberg states derived in this work represent important quantities for future determinations of the adiabatic ionization energies of H2, HD, and D2 at sub-MHz accuracy.

Spectrum of the autoionizing triplet gerade Rydberg states of H2 and its analysis using multichannel quantum-defect theory.

A new spectrum of the autoionizing triplet states of gerade symmetry of H2 has been recorded from the υ″ = 1­4, N″ = 1­3 rovibrational levels of the metastable c 3Π(u)­ state in a supersonic beam. The spectrum consists of overlapping ns and nd Rydberg series with n in the range between 4 and 45 converging to the υ+ = 1­4, N+ = 0­5 levels of the X+ 2Σ(g)+ ground state of H2+. Numerous perturbations caused by s­d and rovibrational channel interactions are revealed in the spectrum and were fully assigned by combining double-resonance experiments and ab initio multichannel quantum-defect theory (MQDT). The energy- and internuclear-distance-dependent eigenquantum-defect parameters of MQDT were derived from available ab initio calculations of the low-lying electronic states of H2 and the ground state of H2(+) and were subsequently refined in a global fit to experimental data. The positions of 552 triplet ns and nd Rydberg levels of H2 (361 of which were measured in the present study) could be reproduced with a root-mean-square deviation of 0.2 cm(­1).

A quantum defect model for the s, p, d, and f Rydberg series of CaF.

We present an improved quantum defect theory model for the "s," "p," "d," and "f" Rydberg series of CaF. The model, which is the result of an exhaustive fit of high-resolution spectroscopic data, parameterizes the electronic structure of the ten ("s"Σ, "p"Σ, "p"Π, "d"Σ, "d"Π, "d"Δ, "f"Σ, "f"Π, "f"Δ, and "f"Φ) Rydberg series of CaF in terms of a set of twenty µ(ll('))(Λ) quantum defect matrix elements and their dependence on both internuclear separation and on the binding energy of the outer electron. Over 1000 rovibronic Rydberg levels belonging to 131 observed electronic states of CaF with n∗ ≥ 5 are included in the fit. The correctness and physical validity of the fit model are assured both by our intuition-guided combinatorial fit strategy and by comparison with R-matrix calculations based on a one-electron effective potential. The power of this quantum defect model lies in its ability to account for the rovibronic energy level structure and nearly all dynamical processes, including structure and dynamics outside of the range of the current observations. Its completeness places CaF at a level of spectroscopic characterization similar to NO and H(2).

Towards measuring the ionisation and dissociation energies of molecular hydrogen with sub-MHz accuracy.

The most precise determination of the ionisation and dissociation energies of molecular hydrogen H2 was carried out recently by measuring three intervals independently: the X --> EF interval, the EF --> n = 54p interval, and the electron binding energy of the n = 54p Rydberg state. The values of the ionisation and dissociation energies obtained for H2, and for HD and D2 in similar measurements, are in agreement with the results of the latest ab initio calculations [Piszczatowski et al., J. Chem. Theory Comput., 2009, 5, 3039; Pachucki and Komasa, Phys. Chem. Chem. Phys., 2010, 12, 9188] within the combined uncertainty limit of 30 MHz (0.001 cm(-1)). We report on a new determination of the electron binding energies of H2 Rydberg states with principal quantum numbers in the range n = 51-64 with a precision of better than 100 kHz using a combination of millimetre-wave spectroscopy and multichannel quantum-defect theory (MQDT). The positions of 33 np (S = 0) Rydberg states of ortho-H2 relative to the position of the reference 51d (N+ = 1, N = 1, G+ = 1/2, G = 1, F = 0) Rydberg state have been determined with a precision and accuracy of 50 kHz. By analysing these positions using MQDT, the electron binding energy of the reference state could be determined to be 42.3009108(14) cm(-1), which represents an improvement by a factor of -7 over the previous value obtained by Osterwalder et al [J. Chem. Phys., 2004, 121, 11810]. Because the electron binding energy of the high-n Rydberg states will ultimately be the limiting factor in our method of determining the ionisation and dissociation energies of molecular hydrogen, this result opens up the possibility of carrying out a new determination of these quantities. By evaluating several schemes for the new measurement, the precision limit is estimated to be 50-100 kHz, approaching the fundamental limit for theoretical values of -10 kHz imposed by the current uncertainty of the proton-to-electron mass ratio.

Communication: The ionization and dissociation energies of HD.

The adiabatic ionization energy [in units of hc, E(i)=124 568.485 81(36) cm(-1)] and the dissociation energy [D(0)=36 405.783 66(36) cm(-1)] of HD have been determined using a hybrid experimental-theoretical method. Experimentally, the wave numbers of the EF(v=0,N=0)→np[X(+)(v(+)=0 and 1, N(+)=0)] and EF(v=0,N=1)→np[X(+)(v(+)=0,N(+)=1)] transitions to singlet Rydberg states were measured by laser spectroscopy and used to validate predictions of the electron binding energies by multichannel quantum defect theory. Adding the transition energies, the electron binding energies and previously reported term energies of the EF state led to a determination of the adiabatic ionization energy of HD and of rovibrational energy spacings in HD(+). Combining these measurements with highly accurate theoretical values of the ionization energies of the one-electron systems H, D, and HD(+) further enabled a new determination of the dissociation energy of HD.

Determination of the ionization and dissociation energies of the deuterium molecule (D(2)).

The transition wave numbers from selected rovibrational levels of the EF (1)Sigma(g) (+)(v=0) state to selected np Rydberg states of ortho- and para-D(2) located below the adiabatic ionization threshold have been measured at a precision better than 10(-3) cm(-1). Adding these wave numbers to the previously determined transition wave numbers from the X (1)Sigma(g) (+)(v=0, N=0,1) states to the EF (1)Sigma(g) (+)(v=0, N=0,1) states of D(2) and to the binding energies of the Rydberg states calculated by multichannel quantum defect theory, the ionization energies of ortho- and para-D(2) are determined to be 124 745.394 07(58) cm(-1) and 124 715.003 77(75) cm(-1), respectively. After re-evaluation of the dissociation energy of D(2) (+) and using the known ionization energy of D, the dissociation energy of D(2) is determined to be 36 748.362 86(68) cm(-1). This result is more precise than previous experimental results by more than one order of magnitude and is in excellent agreement with the most recent theoretical value 36 748.3633(9) cm(-1) [K. Piszczatowski, G. Lach, M. Przybytek et al., J. Chem. Theory Comput. 5, 3039 (2009)]. The ortho-para separation of D(2), i.e., the energy difference between the N=0 and N=1 rotational levels of the X (1)Sigma(g) (+)(v=0) ground state, has been determined to be 59.781 30(95) cm(-1).