Precision spectroscopy of molecular hydrogen

Precision measurements on the hydrogen molecule are of fundamental importance in understanding molecular theory. Comparison of accurate experimental data and theoretical results are used to test the quantum electrodynamics theory and determine physical constants used in the calculation. We review recent advances and perspectives in the precision spectroscopy of molecular hydrogen, representing state-of-the-art molecular spectroscopy methods and cutting-edge high-precision calculations.

The (2-0) R(0) and R(1) transition frequencies of HD determined to a 10-10 relative accuracy by Doppler spectroscopy at 80 K

The Doppler broadened R(0) and R(1) lines of the (2-0) vibrational band of HD have been measured at liquid nitrogen temperature and at pressures of 2 Pa, with a comb referenced continuous-wave cavity ring-down spectrometer set-up. Transition frequencies of 214905335185 kHz and 217105181898 kHz were derived from 33 and 83 recordings, with corresponding root mean squared deviation of 53 and 33 kHz for the R(0) and R(1) transition, respectively. This is the first sub-MHz frequency determination of the R(0) transition frequency and represents a three order of magnitude accuracy improvement compared to literature. The R(1) transition frequency is in very good agreement with previous determinations in saturation regime reported with similar accuracy. To achieve such accuracy, the transition frequency of the (101)-(000) 2₁₁-3₁₂ line of H₂¹⁶O interfering with the R(0) line had to be precisely determined and is reported with a standard error of 100 Hz at 214904329826.49(10) kHz (relative uncertainty of 5 × 10^-13). These measurement sets provide stringent reference values for validating future advances in the theoretical description of the hydrogen (and water) molecule.

Ab Initio Partition Functions and Thermodynamic Quantities for the Molecular Hydrogen Isotopologues

In this work, we calculate the partition functions and thermodynamic quantities of molecular hydrogen isotopologues using the rovibrational energy levels provided by the highly accurate ab initio adiabatic potential energy functions recently determined by Pachucki and Komasa (Pachucki, K.; Komasa, J. J. Chem. Phys. 2014, 141, 224103). The partition functions are calculated by including all bound energy levels of the isotopologues, up to their dissociation limits, plus the quasi-bound levels lying below the centrifugal potential barriers. For the homonuclear isotopologues, H2, D2, and T2, we also determine the partition functions and thermodynamic quantities of the normal mixtures using the statistical treatment recently proposed by Colonna et al. (Colonna, G.; D'Angola, A.; Capitelli, M. Int. J. Hydrogen Energy 2012, 37, 9656) based on the definition of the partition function of the mixture, which avoids inconsistencies in the values of the thermodynamic quantities depending directly on the internal partition function, in the high-temperature limit.

Nonadiabatic, Relativistic, and Leading-Order QED Corrections for Rovibrational Intervals of ^{4}He_{2}^{+} (X ^{2}Σ_{u}^{+})

The rovibrational intervals of the ^{4}He_{2}^{+} molecular ion in its X ^{2}Σ_{u}^{+} ground electronic state are computed by including the nonadiabatic, relativistic, and leading-order quantum-electrodynamics corrections. Good agreement of theory and experiment is observed for the rotational excitation series of the vibrational ground state and the fundamental vibration. The lowest-energy rotational interval is computed to be 70.937 69(10)  cm^{-1} in agreement with the most recently reported experimental value, 70.937 589(23)(60)_{sys}  cm^{-1} [L. Semeria et al., Phys. Rev. Lett. 124, 213001 (2020)PRLTAO0031-900710.1103/PhysRevLett.124.213001].

Precision measurement of the fundamental vibrational frequencies of tritium-bearing hydrogen molecules: T2, DT, HT

High-resolution coherent Raman spectroscopic measurements of all three tritium-containing molecular hydrogen isotopologues T2, DT and HT were performed to determine the ground electronic state fundamental Q-branch (v = 0 → 1, ΔJ = 0) transition frequencies at accuracies of 0.0005 cm-1. An over hundred-fold improvement in accuracy over previous experiments allows the comparison with the latest ab initio calculations in the framework of non-adiabatic perturbation theory including nonrelativisitic, relativisitic and QED contributions. Excellent agreement is found between experiment and theory, thus providing a verification of the validity of the NAPT-framework for these tritiated species. While the transition frequencies were corrected for ac-Stark shifts, the contributions of non-resonant background as well as quantum interference effects between resonant features in the nonlinear spectroscopy were quantitatively investigated, also leading to corrections to the transition frequencies. Methods of saturated CARS with the observation of Lamb dips, as well as the use of continuous-wave radiation for the Stokes frequency were explored, that might pave the way for future higher-accuracy CARS measurements.

Ultrahigh finesse cavity-enhanced spectroscopy for accurate tests of quantum electrodynamics for molecules

We report the most accurate, to the best of our knowledge, measurement of the position of the weak quadrupole S(2) 2-0 line in $ {{\rm D}_2} $D₂. The spectra were collected with a frequency-stabilized cavity ringdown spectrometer (FS-CRDS) with an ultrahigh finesse optical cavity ($ {\cal F} = 637 000 $F=637000) and operating in the frequency-agile, rapid scanning spectroscopy (FARS) mode. Despite working in the Doppler-limited regime, we reached 40 kHz of statistical uncertainty and 161 kHz of absolute accuracy, achieving the highest accuracy for homonuclear isotopologues of molecular hydrogen. The accuracy of our measurement corresponds to the fifth significant digit of the leading term in quantum electrodynamics (QED) correction. We observe $ 2.3\sigma $2.3σ discrepancy with the recent theoretical value.

H2 clumped isotope measurements at natural isotopic abundances

RATIONALE

Molecular hydrogen (H₂ ) is an important gas for atmospheric chemistry, and an indirect greenhouse gas due to its reaction with OH. The isotopic composition of H₂ (δD) has been used to investigate its atmospheric budget; here we add a new observable, the clumped isotopic signature ΔDD, to the tools that can be used to study the global cycle of H₂ .

METHODS

A method for determining ΔDD in H₂ was developed using the high-resolution MAT 253-Ultra isotope ratio mass spectrometer (Thermo Fisher). The HH, HD and DD abundances are quantified at medium resolution (M/ΔM ≈ 6000), which is sufficient for HD⁺ and DD⁺ to be distinguished from H₃ ⁺ and H₂ D⁺ , respectively. The method involves sequential measurement of isotopologues, and DD is measured using an ion counter. For verification, catalytic ΔDD equilibration experiments were performed at temperatures of up to 850°C.

RESULTS

The typical precision obtained for ΔDD is 2-6‰, close to the theoretical counting statistics limit, and adequate for detecting the expected natural variations. Compatibility and medium-term reproducibility are consistent with the precision values. The method was validated using temperature equilibration experiments, which showed a dependence of ΔDD on temperature as expected form theoretical calculations.

CONCLUSIONS

We have established a method for determining ΔDD in H₂ at natural isotopic abundances, with a precision that is adequate for observing the expected variations in atmospheric and other natural H₂ . This method opens the road to new research on the natural H₂ cycle.

Nonrelativistic energy levels of HD

Nonadiabatic exponential functions are employed to solve the four-body Schrödinger equation. Nonrelativistic bound energy levels of the HD molecule are calculated to the relative accuracy of 10^-12-10^-13, which is the first step toward highly accurate prediction of dissociation and transition energies. Such energies, in connection with equally accurate experimental data, will enable refinement of the physical constant and aid the search for deviations caused by yet unknown interactions at the atomic scale.

Fundamental vibration frequency and rotational structure of the first excited vibrational level of the molecular helium ion ( He 2 + )

The term values of the rotational levels of the first excited vibrational state of the electronic ground state of He 2 + with a rotational quantum number N ⁺ ≤ 13 have been determined with an accuracy of 1.2 × 10^-3 cm^-1 (∼35 MHz) by multichannel-quantum-defect-theory-assisted Rydberg spectroscopy of metastable He₂. Comparison of the experimental term values with the most accurate ab initio results for He 2 + available in the literature [W.-C. Tung, M. Pavanello, and L. Adamowicz, J. Chem. Phys. 136, 104309 (2012)] reveals inconsistencies between the theoretical and experimental results that increase with increasing rotational quantum numbers. The fundamental vibrational wavenumber of He 2 + was determined to be 1628.3832(12) cm^-1 by fitting effective molecular constants to the obtained term values.

Relativistic corrections to the ground states of HD and D2 calculated without using the Born-Oppenheimer approximation

The Schrödinger equation for the ground states of the hydrogen molecules HD and D₂ is solved variationally by treating the constituent particles of HD or D₂ on the same footing without assuming the Born-Oppenheimer approximation. The variational basis sets are constructed using Hylleraas coordinates that are traditionally adopted for few-electron atomic systems. The nonrelativistic energy eigenvalues are converged to the level of 10^-6 cm^-1. The leading-order relativistic corrections, including relativistic recoil terms, are calculated rigorously. Together with the higher-order relativistic and quantum electrodynamic corrections obtained by the Pachucki's group [Phys. Rev. A., 2017, 95, 052506; Phys. Rev. Lett., 2018, 120, 153001], we determine the dissociation energy of D₂ to be 36748.36240(28) cm^-1, which agrees with the recent experimental result of Liu et al. [J. Chem. Phys., 2010, 132, 154301] 36748.36286(68) cm^-1. For HD, the dissociation energy determined by us is 36405.78252(27) cm^-1, which deviates from the most accurate experimental result of Sprecher et al. [J. Chem. Phys., 2010, 133, 111102] 36405.78366(36) cm^-1 by about 2σ.

Nonadiabatic Relativistic Correction to the Dissociation Energy of H_{2}, D_{2}, and HD

The relativistic correction to the dissociation energy of H_{2}, D_{2}, and HD molecules has been accurately calculated without expansion in the small electron-nucleus mass ratio. The obtained results indicate the significance of nonadiabatic effects and resolve the discrepancy of theoretical predictions with recent experimental values for H_{2} and D_{2}. While the theoretical accuracy is now significantly improved and is higher than the experimental one, we observe about 3σ discrepancy for the dissociation energy of HD, which requires further investigation.

Relativistic and QED Effects in the Fundamental Vibration of T_{2}

The hydrogen molecule has become a test ground for quantum electrodynamical calculations in molecules. Expanding beyond studies on stable hydrogenic species to the heavier radioactive tritium-bearing molecules, we report on a measurement of the fundamental T_{2} vibrational splitting (v=0→1) for J=0-5 rotational levels. Precision frequency metrology is performed with high-resolution coherent anti-Stokes Raman spectroscopy at an experimental uncertainty of 10-12 MHz, where sub-Doppler saturation features are exploited for the strongest transition. The achieved accuracy corresponds to a 50-fold improvement over a previous measurement, and it allows for the extraction of relativistic and QED contributions to T_{2} transition energies.

Toward a Determination of the Proton-Electron Mass Ratio from the Lamb-Dip Measurement of HD

Precision spectroscopy of the hydrogen molecule is a test ground of quantum electrodynamics (QED), and it may serve for the determination of fundamental constants. Using a comb-locked cavity ring-down spectrometer, for the first time, we observed the Lamb-dip spectrum of the R(1) line in the overtone of hydrogen deuteride (HD). The line position was determined to be 217 105 182.79±0.03_{stat}±0.08_{syst}  MHz (δν/ν=4×10^{-10}), which is the most accurate rovibrational transition ever measured in the ground electronic state of molecular hydrogen. Moreover, from calculations including QED effects up to the order m_{e}α^{6}, we obtained predictions for this R(1) line as well as for the HD dissociation energy, which are less accurate but signaling the importance of the complete treatment of nonadiabatic effects. Provided that the theoretical calculation reaches the same accuracy, the present measurement will lead to a determination of the proton-to-electron mass ratio with a precision of 1.3 parts per billion.

Sub-Doppler Frequency Metrology in HD for Tests of Fundamental Physics

Weak transitions in the (2,0) overtone band of the hydrogen deuteride molecule at λ=1.38  μm were measured in saturated absorption using the technique of noise-immune cavity-enhanced optical heterodyne molecular spectroscopy. Narrow Doppler-free lines were interrogated with a spectroscopy laser locked to a frequency comb laser referenced to an atomic clock to yield transition frequencies [R(1)=217105181895(20)  kHz; R(2)=219042856621(28)  kHz; R(3)=220704304951(28)  kHz] at three orders of magnitude improved accuracy. These benchmark values provide a test of QED in the smallest neutral molecule, and they open up an avenue to resolve the proton radius puzzle, as well as constrain putative fifth forces and extra dimensions.

Polarizability tensor invariants of H2, HD, and D2

We report an exhaustive compilation of wavelength-dependent matrix elements over the mean polarizability (α¯) and polarizability anisotropy (γ) operators for the rovibrational states of the H₂, HD, and D₂ molecules together with an accompanying computer program for their evaluation. The matrix elements can be readily evaluated using the provided codes for rovibrational states with J = 0-15 and v = 0-4 and for any laser wavelengths in the interval 182.25-1320.6 nm corresponding to popular, commercially available lasers. The presented results substantially extend the scope of the data available in the literature, both in respect of the rovibrational transitions analyzed and the range of covered laser frequencies. The presented detailed tabulation of accurate polarizability tensor invariants is essential for successful realization of our main long-term goal: developing a universal standard for determining absolute Raman cross sections and absolute Raman intensities in experimental Rayleigh and Raman scattering studies of molecules.

Pair Potential with Submillikelvin Uncertainties and Nonadiabatic Treatment of the Halo State of the Helium Dimer

The pair potential for helium is computed with accuracy improved by an order of magnitude relative to the best previous determination. For the well region, its uncertainties are now below 1 millikelvin. The main improvement is due to the use of explicitly correlated wave functions at the nonrelativistic Born-Oppenheimer (BO) level of theory. The diagonal BO and the relativistic corrections are obtained from large full configuration interaction calculations. Nonadiabatic perturbation theory is used to predict the properties of the halo state of the helium dimer. Its binding energy and the average value of the interatomic distance are found to be 138.9(5) neV and 47.13(8) Å. The binding energy agrees with its first experimental determination of 151.9(13.3) neV [Zeller et al., Proc. Natl. Acad. Sci. U.S.A. 113, 14651 (2016)PNASA60027-842410.1073/pnas.1610688113].

Precision Spectroscopy in Cold Molecules: The Lowest Rotational Interval of He_{2}^{+} and Metastable He_{2}

Multistage Zeeman deceleration was used to generate a slow, dense beam of translationally cold He_{2} molecules in the metastable a ^{3}Σ_{u}^{+} state. Precision measurements of the Rydberg spectrum of these molecules at high values of the principal quantum number n have been carried out. The spin-rotational state selectivity of the Zeeman-deceleration process was exploited to reduce the spectral congestion, minimize residual Doppler shifts, resolve the Rydberg series around n=200 and assign their fine structure. The ionization energy of metastable He_{2} and the lowest rotational interval of the X^{+} ^{2}Σ_{u}^{+} (ν^{+}=0) ground state of ^{4}He_{2}^{+} have been determined with unprecedented precision and accuracy by Rydberg-series extrapolation. Comparison with ab initio predictions of the rotational energy level structure of ^{4}He_{2}^{+} [W.-C. Tung, M. Pavanello, and L. Adamowicz, J. Chem. Phys. 136, 104309 (2012)] enabled us to quantify the magnitude of relativistic and quantum-electrodynamics contributions to the fundamental rotational interval of He_{2}^{+}.

Modelling non-adiabatic effects in H₃⁺: solution of the rovibrational Schrödinger equation with motion-dependent masses and mass surfaces

Introducing different rotational and vibrational masses in the nuclear-motion Hamiltonian is a simple phenomenological way to model rovibrational non-adiabaticity. It is shown on the example of the molecular ion H3(+), for which a global adiabatic potential energy surface accurate to better than 0.1 cm(-1) exists [M. Pavanello, L. Adamowicz, A. Alijah, N. F. Zobov, I. I. Mizus, O. L. Polyansky, J. Tennyson, T. Szidarovszky, A. G. Császár, M. Berg et al., Phys. Rev. Lett. 108, 023002 (2012)], that the motion-dependent mass concept yields much more accurate rovibrational energy levels but, unusually, the results are dependent upon the choice of the embedding of the molecule-fixed frame. Correct degeneracies and an improved agreement with experimental data are obtained if an Eckart embedding corresponding to a reference structure of D(3h) point-group symmetry is employed. The vibrational mass of the proton in H3(+) is optimized by minimizing the root-mean-square (rms) deviation between the computed and recent high-accuracy experimental transitions. The best vibrational mass obtained is larger than the nuclear mass of the proton by approximately one third of an electron mass, m(opt,p)((v))=m(nuc,p)+0.31224m(e). This optimized vibrational mass, along with a nuclear rotational mass, reduces the rms deviation of the experimental and computed rovibrational transitions by an order of magnitude. Finally, it is shown that an extension of the algorithm allowing the use of motion-dependent masses can deal with coordinate-dependent mass surfaces in the rovibrational Hamiltonian, as well.

Observation of g/u-symmetry mixing in the high-n Rydberg states of HD

The structure and dynamics of high-n Rydberg states belonging to series converging to the (v(+) = 0, N(+) = 0-2) levels of the X(+) (2)Σ(g)(+)electronic ground state of HD(+) were studied by high-resolution spectroscopy from the GK (1)Σ(g)(+) (v = 1, N = 1) state under field-free conditions. Three effects of g/u-symmetry breaking were detected: (i) Single-photon transitions from the GK (v = 1, N = 1) state of gerade symmetry to the 30d21 and 31g22 Rydberg states of gerade symmetry were observed after careful compensation of the stray electric fields. (ii) The singlet 61p12 Rydberg state of ungerade symmetry was found to autoionize to the N(+) = 0, ℓ = 2 ionization continuum of gerade symmetry with a lifetime of 77(10) ns. (iii) Shifts of up to 20 MHz induced by g/u-symmetry mixing were measured for members of the np11 Rydberg series which lie close to nd21 Rydberg states. These observations were analyzed in the framework of multichannel quantum-defect theory. From the observed level shifts, the off-diagonal eigenquantum-defect element μ(pd) of singlet-π symmetry was determined to be 0.0023(3) and the corresponding autoionization dynamics could be characterized. The ionization energy of the GK (v = 1, N = 1) state of HD was determined to be 12 710.544 23(10) cm(-1).

Charge asymmetry in the rovibrationally excited HD molecule

The recently developed method for performing all-particle non-Born-Oppenheimer variational calculations on diatomic molecular systems excited to the first excited rotational state and simultaneously vibrationally excited is employed to study the charge asymmetry and the level lifetimes of the HD molecule. The method uses all-particle explicitly correlated Gaussian functions. The nonlinear parameters of the Gaussians are optimized with the aid of the analytical energy gradient determined with respect to these parameters.

Core-mass nonadiabatic corrections to molecules: H2, H2+, and isotopologues

For high-precision calculations of rovibrational states of light molecules, it is essential to include non-adiabatic corrections. In the absence of crossings of potential energy surfaces, they can be incorporated in a single surface picture through coordinate-dependent vibrational and rotational reduced masses. We present a compact method for their evaluation and relate in particular the vibrational mass to a well defined nuclear core mass derived from a Mulliken analysis of the electronic density. For the rotational mass we propose a simple, but very effective parametrization. The use of these masses in the nuclear Schrödinger equation yields numerical data for the corrections of a much higher quality than can be obtained with optimized constant masses, typically better than 0.1 cm(-1). We demonstrate the method for H(2), H(2)(+), and singly deuterated isotopologues. Isotopic asymmetry does not present any particular difficulty. Generalization to polyatomic molecules is straightforward.

Benchmarking Experimental and Computational Thermochemical Data: A Case Study of the Butane Conformers

Due to its crucial importance, numerous studies have been conducted to determine the enthalpy difference between the conformers of butane. However, it is shown here that the most reliable experimental values are biased due to the statistical model utilized during the evaluation of the raw experimental data. In this study, using the appropriate statistical model, both the experimental expectation values and the associated uncertainties are revised. For the 133-196 and 223-297 K temperature ranges, 668 ± 20 and 653 ± 125 cal mol(-1), respectively, are recommended as reference values. Furthermore, to show that present-day quantum chemistry is a favorable alternative to experimental techniques in the determination of enthalpy differences of conformers, a focal-point analysis, based on coupled-cluster electronic structure computations, has been performed that included contributions of up to perturbative quadruple excitations as well as small correction terms beyond the Born-Oppenheimer and nonrelativistic approximations. For the 133-196 and 223-297 K temperature ranges, in exceptional agreement with the corresponding revised experimental data, our computations yielded 668 ± 3 and 650 ± 6 cal mol(-1), respectively. The most reliable enthalpy difference values for 0 and 298.15 K are also provided by the computational approach, 680.9 ± 2.5 and 647.4 ± 7.0 cal mol(-1), respectively.

Precision measurements and computations of transition energies in rotationally cold triatomic hydrogen ions up to the midvisible spectral range

First-principles computations and experimental measurements of transition energies are carried out for vibrational overtone lines of the triatomic hydrogen ion H(3)(+) corresponding to floppy vibrations high above the barrier to linearity. Action spectroscopy is improved to detect extremely weak visible-light spectral lines on cold trapped H(3)(+) ions. A highly accurate potential surface is obtained from variational calculations using explicitly correlated Gaussian wave function expansions. After nonadiabatic corrections, the floppy H(3)(+) vibrational spectrum is reproduced at the 0.1 cm(-1) level up to 16600 cm(-1).

QED effects in molecules: test on rotational quantum states of H2

Quantum electrodynamic effects have been systematically tested in the progression of rotational quantum states in the X 1Σ(g)(+), v=0 vibronic ground state of molecular hydrogen. High-precision Doppler-free spectroscopy of the EF 1Σ(g)(+)-X 1Σ(g)(+) (0,0) band was performed with 0.005  cm(-1) accuracy on rotationally hot H2 (with rotational quantum states J up to 16). QED and relativistic contributions to rotational level energies as high as 0.13  cm(-1) are extracted, and are in perfect agreement with recent calculations of QED and high-order relativistic effects for the H2 ground state.