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Journal of Chemical Physics / Volume 134 / Issue 11 / ARTICLES / Gas Phase Dynamics and Structure: Spectroscopy, Molecular Interactions, Scattering, and Photochemistry
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J. Chem. Phys. 134, 114313 (2011); http://dx.doi.org/10.1063/1.3565967 (21 pages)

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

Jeffrey J. Kay1, Stephen L. Coy1, Bryan M. Wong1, Christian Jungen2,3, and Robert W. Field1

1Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
2Laboratoire Aimé Cotton du CNRS, Université de Paris Sud, Bâtiment 505, F-91405 Orsay, France
3Department of Physics and Astronomy, University College London, London WC1E 6BT, United Kingdom

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(Received 20 September 2010; accepted 18 February 2011; published online 18 March 2011; corrected 30 January 2012)

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 μℓℓ′(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 H2.

© 2011 American Institute of Physics

Article Outline

  1. INTRODUCTION
  2. EXPERIMENT
  3. THEORY
  4. RESULTS AND DISCUSSION
    1. Application to CaF
    2. Fit procedure
      1. Initialization and convergence using mathℓℓ′(Λ)|Re+ and ∂mathℓℓ′(Λ)/∂R
      2. Generation of estimates of ∂mathℓℓ′(Λ)/∂ɛ
      3. Convergence using mathℓℓ′(Λ)|Re+ , ∂mathℓℓ′(Λ)/∂R , and ∂mathℓℓ′(Λ)/∂ɛ
      4. Refinement by addition of second derivatives: ∂2mathℓℓ′(Λ)/∂R2 , ∂2mathℓℓ′(Λ)/∂ɛ2 , and ∂2mathℓℓ′(Λ)/∂ɛ∂R
    3. Quantum defect matrices and quality of fit
    4. R -matrix estimates of quantum defect matrix elements
  5. CONCLUSIONS

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KEYWORDS and PACS

Keywords

binding energy, calcium compounds, molecular configurations, quantum theory, rotational-vibrational states, Rydberg states

PACS

  • 33.15.Ry

    Ionization potentials, electron affinities, molecular core binding energy

  • 33.15.Mt

    Rotation, vibration, and vibration-rotation constants

  • 33.20.Vq

    Vibration-rotation analysis

  • 33.15.Bh

    General molecular conformation and symmetry; stereochemistry

ARTICLE DATA

Digital Object Identifier
http://dx.doi.org/10.1063/1.3565967

PUBLICATION DATA

ISSN

0021-9606 (print)  
1089-7690 (online)

Publisher

$abstract.society.value is a Member of CrossRef American Institute of Physics

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Figures (click on thumbnails to view enlargements)

FIG.1
Comparison of selected observed and calculated energy levels near n* = 5.0, where vibronic states tend to be well-separated. For visual clarity, the reduced energy E − B+N(N + 1) is plotted against N(N + 1) for each energy level. Filled circles indicate calculated energy levels and connected open circles indicate observed energy levels. Many levels with n* < 6.0 are fitted to within 1 cm−1, and most to within 5 cm−1.

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FIG.2
Comparison of selected observed and calculated energy levels for vibrationally excited levels with low n*. For visual clarity, the reduced energy E − B+N(N + 1) is plotted against N(N + 1) for each energy level. Many levels with n* < 6.0 are fitted to within 1 cm−1, and most to within 5 cm−1.

FIG.2 Download High Resolution Image (.zip file) | Export Figure to PowerPoint

FIG.3
Comparison of selected observed and calculated energy levels in the vicinity of n* = 7.0. For visual clarity, the reduced energy E − B+N(N + 1) is plotted against N(N + 1) for each energy level. Vibronic states at this energy are interleaved. Here, the classical period of electronic motion [proportional to (n*)3] is approximately equal to the classical period of vibrational motion. Vibronic perturbations are frequent.

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FIG.4
Example of a strong vibronic (homogeneous) perturbation. In the absence of the perturbation, the 7.36 “p” Π v = 0 and 6.36 “p” Π v = 1 levels are nearly degenerate. The perturbation causes a ∼45 cm−1 splitting of the levels and complete mixing of the two zero-order wavefunctions.

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FIG.5
Comparison of selected observed and calculated energy levels in the vicinity of n* = 14.0. At this energy, the electronic energy level spacing is much smaller than the vibrational spacing, but still larger than the rotational spacing of the ion-core energy levels. Vibronic perturbations are uncommon, but rotational (inhomogeneous) perturbations become increasingly frequent.

FIG.5 Download High Resolution Image (.zip file) | Export Figure to PowerPoint

FIG.6
Quality of fit in the 14f complex. A rotational perturbation between the 14f Π− and 14.14 “d” Δ− states gives rise to the avoided crossing at the top of the figure.

FIG.6 Download High Resolution Image (.zip file) | Export Figure to PowerPoint

FIG.7
Quality of fit in the n* = 16.5 – 17.5 region. Above n* ∼ 16, rotational interactions are ubiquitous and quite strong, causing the disappearance of regular patterns which is evident here.

FIG.7 Download High Resolution Image (.zip file) | Export Figure to PowerPoint

FIG.8
R-dependence of MQDT-fitted and R-matrix calculated eigenquantum defects for (a) Σ and (b) Π series with E = -0.020 Ry (n* ≈ 7.0) and E = −0.012 Ry (n* ≈ 9.0), respectively. R = 3.54 a0, is the equilibrium internuclear separation of the ion core.

FIG.8 Download High Resolution Image (.zip file) | Export Figure to PowerPoint

FIG.9
Energy dependence of MQDT-fitted and R-matrix calculated Σ and Π series eigenquantum defects at the equilibrium internuclear separation, R = 3.54 a0. Energy is in Rydberg units.

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FIG.10
Comparison of energy dependence of Σ series MQDT-fitted and R-matrix calculated math matrix elements, at the equilibrium internuclear separation, R = 3.54 a0. Energy is in Rydberg units. The calculated math matrix elements have been adjusted as discussed in Appendix C to allow direct comparison with the fitted matrix elements.

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FIG.11
R-dependence of MQDT-fitted and R-matrix calculated math matrix elements for Σ series, E = –0.02 Ry (n* ≈ 7.0). Trends with R show some differences from the experimental result away from the equilibrium R. (Also see Appendix C.) The calculated math matrix elements have been adjusted as discussed in Appendix C to allow direct comparison with the fitted matrix elements.

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FIG.12
Comparison of R-dependence of MQDT-fitted and R-matrix calculated math matrix elements for Π series, E = –0.012 Ry (n* ≈ 7.0). Trends with R show some differences from the experimental values away from Re. (Also see Appendix C.) The calculated math matrix elements have been adjusted as discussed in Appendix C to allow direct comparison with the fitted matrix elements.

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FIG.13
Calculated R-dependence of higher-ℓ mixing in Σ and Π states of dominant f character, at E = –0.02 Ry. The calculation predicts mixing outside of the experimentally fitted s, p, d, f block dominantly to g character, but also to h. These mixings enhance experimental access to non-penetrating states, as reported in Kay et al. (Ref. 31).

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Supplemental Files (EPAPS)

Tables

Table I. math quantum defect matrix element values and derivatives obtained from fits to CaF Σ, Π, Δ, and Φ states. Uncertainties are indicated in parentheses. If no numerical value is given, the parameter has been held fixed at zero.

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Table II. UCF(R = 3.54a0) matrix for Σ symmetry.

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Table III. UCF(R = 3.54a0) matrix for Π symmetry.

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