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Quantum computation solves a half-century-old enigma: Elusive vibrational states of magnesium dimer found

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Science Advances  03 Apr 2020:
Vol. 6, no. 14, eaay4058
DOI: 10.1126/sciadv.aay4058
  • Fig. 1 The wave functions of the high-lying, purely vibrational, states of 24Mg2 and the underlying X1Σg+ potential.

    The last experimentally observed v″ = 13 level is marked in blue, the predicted v″ = 14 to 18 levels are marked in green, and the ab initio X1Σg+ PEC obtained in this study is marked by a long-dashed black line. The inset is a Birge-Sponer plot comparing the rotationless G(v″ + 1) − G(v″) energy differences as functions of v″ + ½ obtained in this work (black circles) with their experimentally derived counterparts (red open squares) based on the data reported in (12) (v″ = 0 to 12) and (17) (v″ = 13; cf. also Table 1). The red solid line is a linear fit of the experimental points.

  • Fig. 2 Schematics of the pump, X1Σg+(v″ = 5, J″ = 10) → A1Σu+(v′ = 3, J′ = 11), and fluorescence, A1Σu+(v′ = 3, J′ = 11) → X1Σg+(v″, J″ = 10,12), processes resulting in the LIF spectrum for 24Mg2 shown in figure 3 of (20).

    The X1Σg+ and A1Σu+ PECs and the corresponding X1Σg+(v″ = 5, J″ = 10) and A1Σu+(v′ = 3, J′ = 11) rovibrational wave functions were calculated in this work. The A1Σu+ PEC was shifted to match the experimentally determined adiabatic electronic excitation energy Te of 26,068.9 cm−1 (21) (see Materials and Methods for the details).

  • Fig. 3 The A1Σu+(v′ = 3, J′ = 11) → X1Σg+(v″, J″ = 10,12) LIF spectrum of 24Mg2.

    (A) Comparison of the experimental A1Σu+(v′ = 3, J′ = 11) → X1Σg+(v″, J″ = 10,12) fluorescence progression [black solid lines; adapted from figure 3 of (20) with the permission of AIP Publishing] with its ab initio counterpart obtained in this work (red dashed lines). The theoretical line intensities were normalized such that the tallest peaks in the calculated and experimental spectra corresponding to the v″ = 5 P12 line match. (B) Magnification of the low-energy region of the LIF spectrum shown in (A), with red solid lines representing the calculated transitions. The blue arrows originating from the v″ = 13 label indicate the location of the experimentally observed v″ = 13 P12/R10 doublet. The blue arrows originating from the v″ = 14 and 15 labels point to the most probable locations of the corresponding P12/R10 doublets. Spectral lines involving v″ = 16 and 17 are buried in the noise (see also Table 3).

  • Table 1 Comparison of the ab initio (Calc.) and experimentally derived (Expt.) rovibrational G(v″, J″) energies for selected values of J″ characterizing 24Mg2 in the ground electronic state (in reciprocal centimeter), along with the corresponding dissociation energies De (in reciprocal centimeter) and equilibrium bond lengths re (in angstrom).

    The G(v″, J″) energies calculated using the ab initio X1Σg+ PEC defined by Eq. 1 are reported as errors relative to experiment, whereas De and re are the actual values of these quantities. If the experimental G(v″, J″) energies are not available, we provide their calculated values in square brackets. Quasi-bound rovibrational levels are given in italics. Horizontal bars indicate term values not supported by the X1Σg+ PEC.

    vG(v″, J″ = 0)G(v″, J″ = 20)G(v″, J″ = 40)G(v″, J″ = 60)G(v″, J″ = 80)
    Calc.Expt.*Calc.Expt.Calc.Expt.Calc.Expt.Calc.Expt.
    00.025.2−0.263.3−0.4171.2−0.9340.4−1.8552.8
    1−0.273.0−0.4109.7−0.7213.1−1.2374.6−2.2573.2
    2−0.5117.8−0.7153.0−1.0252.0−1.6405.4[585.0]
    3−0.7159.4−1.0193.2−1.3287.7−1.9432.9
    4−0.9198.0−1.3230.3−1.6320.3−2.1456.7
    5−1.1233.6−1.5264.4−1.8349.7−2.1476.5
    6−1.2266.2−1.7295.5−1.9375.9−1.7491.7
    7−1.3295.8−1.8323.6−1.9398.8
    8−1.4322.5−1.7348.5−1.7418.1
    9−1.4346.2−1.6370.3−1.4433.9
    10−1.3366.8−1.4389.0[444.5]
    11−1.2384.4−1.2404.4[451.6]
    12−0.9398.8−0.9416.6
    13−0.7410.3−0.5425.5
    14[418.4][431.1]
    15[424.6]
    16[428.4]
    17[430.4]
    18[431.2]
    De431.4430.472
    re3.8933.89039

    *Experimentally derived values for v″ = 0 to 12 taken from (12). The v″ = 13 value is calculated as G(v″ = 13, J″ = 14) − 210B(v″ = 13, J″ = 14) with the information about G(v″ = 13, J″ = 14) and B(v″ = 13, J″ = 14) taken from (17).

    †Experimentally derived values taken from the supplementary material of (21).

    ‡Experimentally derived values taken from (20, 21) assuming the X-representation of the X1Σg+ potential developed in (20).

    • Table 2 Comparison of the rovibrational G(v″, J″) energies obtained using the ab initio X1Σg+ PEC defined by Eq. 1 (Calc.) and its X-representation counterpart constructed in (20) (X-rep.) for selected values of J″ characterizing 24Mg2 in the ground electronic state (in reciprocal centimeter), along with the corresponding dissociation energies De (in reciprocal centimeter) and equilibrium bond lengths re (in angstrom).

      The G(v″, J″) energies calculated using the ab initio X1Σg+ PEC are reported as errors relative to the X-representation data, whereas De and re are the actual values of these quantities. If a given G(v″, J″) state corresponding to our ab initio X1Σg+ PEC is not supported by the X-representation potential of (20), we provide its energy in square brackets. Quasi-bound rovibrational levels are given in italics. Horizontal bars indicate term values not supported by the X1Σg+ PEC.

      vG(v″, J″ = 0)G(v″, J″ = 20)G(v″, J″ = 40)G(v″, J″ = 60)G(v″, J″ = 80)
      Calc.X-rep.Calc.X-rep.Calc.X-rep.Calc.X-rep.Calc.X-rep.
      0−0.125.2−0.263.3−0.4171.2−0.9340.4−1.8552.8
      1−0.373.1−0.4109.7−0.7213.1−1.2374.6−2.2573.2
      2−0.6117.9−0.7153.0−1.0252.0−1.6405.4[585.0]
      3−0.9159.6−1.0193.2−1.3287.7−1.9432.9
      4−1.1198.2−1.2230.3−1.6320.3−2.1456.7
      5−1.4233.9−1.5264.4−1.8349.7−2.1476.5
      6−1.5266.5−1.6295.5−1.9375.9−1.7491.7
      7−1.7296.2−1.7323.5−1.9398.8
      8−1.7322.8−1.7348.5−1.7418.1
      9−1.6346.4−1.6370.3−1.4433.8
      10−1.5367.0−1.4389.0−1.0445.5
      11−1.3384.5−1.2404.4[451.6]
      12−1.0399.0−0.9416.6
      13−0.7410.4−0.5425.5
      14−0.5418.9−0.2431.2
      15−0.2424.7
      160.2428.3
      170.5429.9
      180.8430.4
      De431.4430.472
      re3.8933.89039
    • Table 3 Comparison of the theoretical line positions of the A1Σu+(v′ = 3, J′ = 11) → X1Σg+(v″, J″ = 10,12) fluorescence progression in the LIF spectrum of 24Mg2 calculated in this work with experiment.

      All line positions are in reciprocal centimeter. The available experimental values are the actual line positions, whereas our calculated results are errors relative to experiment. If the experimentally determined line positions are not available, we provide their calculated values in square brackets. Horizontal bars indicate term values not supported by the X1Σg+ PEC.

      vP12R10
      Calc.Expt.*Calc.Expt.*
      0−1.526,701.9−1.526,706.0
      1−1.226,654.5−1.326,658.5
      2−1.026,610.3−1.026,614.1
      3−0.726,569.2−0.726,572.8
      4−0.426,531.1−0.526,534.6
      5−0.226,496.0−0.226,499.3
      60.026,463.9[26,467.1]
      70.126,434.90.126,437.9
      80.126,408.80.126,411.7
      90.026,385.90.026,388.5
      10−0.226,366.0−0.226,368.4
      11−0.426,349.2−0.426,351.4
      12−0.726,335.6−0.626,337.5
      13−1.026,325.0−0.926,326.7
      14[26,316.2][26,317.7]
      15[26,311.1][26,312.2]
      16[26,308.4][26,309.1]
      17[26,308.0]
      18

      *Differences between the experimental X1Σg+(v″, J″ = 10,12) and A1Σu+(v′ = 3, J′ = 11) term values reported in the supplementary material of (21) (see the Supplementary Materials), unless stated otherwise.

      †The X1Σg+(v″ = 5, J″ = 10) → A1Σu+(v′ = 3, J′ = 11) pump frequency reported in figure 3 of (20).

      Supplementary Materials

      • Supplementary Materials

        Quantum computation solves a half-century-old enigma: Elusive vibrational states of magnesium dimer found

        Stephen H. Yuwono, Ilias Magoulas, Piotr Piecuch

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        • Sections S1 to S4
        • Figs. S1 to S3
        • Tables S1 to S5
        • Legends for data files S1 and S2

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