^{1}, J. Bai

^{1}, E. Ahmed

^{1}, A. M. Lyyra

^{1}, S. Kotochigova

^{1}, A. J. Ross

^{2}, C. Effantin

^{2}, P. Zalicki

^{3}, J. Vigué

^{4}, G. Chawla

^{5}, R. W. Field

^{5}, T.-J. Whang

^{6}, W. C. Stwalley

^{7}, H. Knöckel

^{8}, E. Tiemann

^{8}, J. Shang

^{9}, L. Li

^{9}and T. Bergeman

^{10}

### Abstract

The lowest electronically excited states of are of interest as intermediaries in the excitation of higher states and in the development of methods for producing cold molecules. We have compiled previously obtained spectroscopic data on the and states of from about 20 sources, both published and unpublished, together with new sub-Doppler linewidth measurements of about 15 000 transitions using polarizationspectroscopy. We also present new *ab initio* results for the diagonal and off-diagonal spin-orbit functions. The discrete variable representation is used in conjunction with Hund’s case potentials plus spin-orbit effects to model data extending from to very close to the limit. Empirical estimates of the spin-orbit functions agree well with the *ab initio* functions for the accessible values of . The potential function for the state includes an exchange potential for atoms, with a fitted coefficient somewhat larger than the predicted value. Observed and calculated term values are presented in an auxiliary (EPAPS) file as a database for future studies on .

The authors are very grateful to M. Aubert-Frécon for communications regarding the exchange potential, to W. Zemke for discussions on matching the short-range and long-range potentials, and to B. Beser (a graduate student at Temple University) for detecting a flaw in our state potential at one point. The authors are pleased to acknowledge support by NSF (Temple University, Stony Brook, and MIT), AOR (Temple and SB) and ONR (SB).

I. INTRODUCTION

II. THE EXPERIMENTAL DATA

A. Data previously published

B. Unpublished data

III. FITTED POTENTIALS AND SPIN-ORBIT FUNCTIONS

A. Molecular Hamiltonian

B. The fitting process

C. The potential functions

D. Spin-orbit functions

E. Calculated term values

IV. CONCLUSION

### Key Topics

- Sodium
- 45.0
- Ab initio calculations
- 28.0
- Dissociation
- 19.0
- Calibration
- 16.0
- Spin orbit interactions
- 16.0

## Figures

Summary of the data used in this study. The observed term values are sorted according to the largest component in the calculated eigenvector.

Summary of the data used in this study. The observed term values are sorted according to the largest component in the calculated eigenvector.

Adiabatic potentials for *ungerade *states in the energy region of this study, obtained by diagonalizing the DVR matrix for , for each value of . Vibrational quantum numbers are shown for the and states. The top plot shows the lower levels, with an inset showing the avoided crossing region. The bottom plot, also with an inset, shows levels and turning points closer to the dissociation limit. Eigenvalue calculations, however, used diabatic potentials and spin-orbit coupling functions.

Adiabatic potentials for *ungerade *states in the energy region of this study, obtained by diagonalizing the DVR matrix for , for each value of . Vibrational quantum numbers are shown for the and states. The top plot shows the lower levels, with an inset showing the avoided crossing region. The bottom plot, also with an inset, shows levels and turning points closer to the dissociation limit. Eigenvalue calculations, however, used diabatic potentials and spin-orbit coupling functions.

A schematic of the experimental setup for polarization spectroscopy. Lateral detection of the total fluorescence records a Doppler-limited laser excitation spectrum.

A schematic of the experimental setup for polarization spectroscopy. Lateral detection of the total fluorescence records a Doppler-limited laser excitation spectrum.

A typical region of the polarization spectrum showing (top) the vernier etalon signal (VET) signal used to check that the laser scans continuously, (middle) the LES signal for calibration, (bottom) the Doppler-limited LES signal, and the polarization spectroscopy signal, which exhibits a sub-Doppler width.

A typical region of the polarization spectrum showing (top) the vernier etalon signal (VET) signal used to check that the laser scans continuously, (middle) the LES signal for calibration, (bottom) the Doppler-limited LES signal, and the polarization spectroscopy signal, which exhibits a sub-Doppler width.

The range of the data for the and states from several of the individual data sets. The numbers in parentheses refer to the data source numbers listed in the first column of Table I.

The range of the data for the and states from several of the individual data sets. The numbers in parentheses refer to the data source numbers listed in the first column of Table I.

This plot shows relative band intensities, calculated from the product of the vibrational overlap, , including the transition moment from Ref. 66, times the Boltzmann factor, . Here, , .

This plot shows relative band intensities, calculated from the product of the vibrational overlap, , including the transition moment from Ref. 66, times the Boltzmann factor, . Here, , .

Residuals from (a) the data from Refs. 23 (dots) and 21 (crosses), (b) the new polarization spectroscopy data, and (c) all other data sets included in the fit. The energy in (a) is scaled as to expand the region near the dissociation limit, , so as to show systematic effects due to hyperfine structure above . Vibrational quantum numbers (smaller dots in a nearly straight line) are plotted with reference to the right axis. Data for (b) were selected to be within of the energy calculated from the fitted parameters.

Residuals from (a) the data from Refs. 23 (dots) and 21 (crosses), (b) the new polarization spectroscopy data, and (c) all other data sets included in the fit. The energy in (a) is scaled as to expand the region near the dissociation limit, , so as to show systematic effects due to hyperfine structure above . Vibrational quantum numbers (smaller dots in a nearly straight line) are plotted with reference to the right axis. Data for (b) were selected to be within of the energy calculated from the fitted parameters.

(a) The fitted short-range (dashed) and long-range (solid line) potentials near the transition point . (b) The difference between the short-range and long-range potentials (solid line), the theoretical exchange potential from Ref. 71 (dot-dashed line), and the fitted exchange potential (dashed line). In (a) and (b) the vertical bar denotes the value of the transition point, .

(a) The fitted short-range (dashed) and long-range (solid line) potentials near the transition point . (b) The difference between the short-range and long-range potentials (solid line), the theoretical exchange potential from Ref. 71 (dot-dashed line), and the fitted exchange potential (dashed line). In (a) and (b) the vertical bar denotes the value of the transition point, .

(a) The sum of the contributions to as a function of , together with plots of the damping corrections to , , and , labeled 3, 6, and , respectively. (b) The individual components of plotted on a logarithmic scale.

(a) The sum of the contributions to as a function of , together with plots of the damping corrections to , , and , labeled 3, 6, and , respectively. (b) The individual components of plotted on a logarithmic scale.

(a) The monotonically decreasing functions represent the logarithm of three estimates of for the state. From top to bottom, these are the experimental function, (dotted line), an *ab initio* function (from SK, dashed line), and the analytic exchange function from Ref. 71 (solid line). The peaked functions are the ratios of the first two to the last, plotted on the linear scale (right axis). (b) The logarithm of for the state obtained from one-half the difference , together with the ratio of this function to the corresponding analytic exchange function from Ref. 71. The inset in (b) shows the two potentials ( vs Å).

(a) The monotonically decreasing functions represent the logarithm of three estimates of for the state. From top to bottom, these are the experimental function, (dotted line), an *ab initio* function (from SK, dashed line), and the analytic exchange function from Ref. 71 (solid line). The peaked functions are the ratios of the first two to the last, plotted on the linear scale (right axis). (b) The logarithm of for the state obtained from one-half the difference , together with the ratio of this function to the corresponding analytic exchange function from Ref. 71. The inset in (b) shows the two potentials ( vs Å).

Diagonal state spin-orbit function . The dashed line denotes results of a fit to a Taylor series expansion about that is made possible by the range of values of as a function of vibrational level as indicated below the curve. The solid line denotes the best fit to a Morse function [Eq. (7)] adjusted to match the Taylor series expansion as closely as possible. The filled circles and connecting line denote *ab initio* results. The inset shows an overview of the asymptotic region.

Diagonal state spin-orbit function . The dashed line denotes results of a fit to a Taylor series expansion about that is made possible by the range of values of as a function of vibrational level as indicated below the curve. The solid line denotes the best fit to a Morse function [Eq. (7)] adjusted to match the Taylor series expansion as closely as possible. The filled circles and connecting line denote *ab initio* results. The inset shows an overview of the asymptotic region.

Off-diagonal spin-orbit coupling functions. The solid-line curves show various fitted Morse function expressions. Each of these curves gives a comparably good fit to the experimental data. The inset shows that they cross within a small region near the crossing point of the and state potentials. The connected squares denote the *ab initio* results of SK which were used in the final fit.

Off-diagonal spin-orbit coupling functions. The solid-line curves show various fitted Morse function expressions. Each of these curves gives a comparably good fit to the experimental data. The inset shows that they cross within a small region near the crossing point of the and state potentials. The connected squares denote the *ab initio* results of SK which were used in the final fit.

Calculated and observed term values showing the extent of the data in the region of perturbations for low levels of the state.

Calculated and observed term values showing the extent of the data in the region of perturbations for low levels of the state.

Calculated (circles) and observed (crosses) term values for of the state with the deperturbed values subtracted. The quality of the fit is shown on an expanded scale relative to the previous figure. Divergences appear at the perturbation crossings.

Calculated (circles) and observed (crosses) term values for of the state with the deperturbed values subtracted. The quality of the fit is shown on an expanded scale relative to the previous figure. Divergences appear at the perturbation crossings.

Calculated and observed term values over a larger region. In this figure, the filled dots represent experimental data from any of the sources.

Calculated and observed term values over a larger region. In this figure, the filled dots represent experimental data from any of the sources.

## Tables

Sources of the data used in the present work. The first number in parentheses after or denotes vibrational numbers, after the semicolon, the rotational quantum numbers. PW, present work; Abs. Spec., absorption spectroscopy; LS, laser spectroscopy; Mod. Pop. Spec., modulated population spectroscopy; LES, laser excitation specroscopy; Mod. Gain, modulated gain; Pol. Spec., polarization spectroscopy; AOTR, all-optical triple resonance; DR, double resonance; PFAOTR, perturbation-facilitated all-optical triple resonance; FT, Fourier transform; PFOODR, perturbation-facilitated optical-optical double resonance.

Sources of the data used in the present work. The first number in parentheses after or denotes vibrational numbers, after the semicolon, the rotational quantum numbers. PW, present work; Abs. Spec., absorption spectroscopy; LS, laser spectroscopy; Mod. Pop. Spec., modulated population spectroscopy; LES, laser excitation specroscopy; Mod. Gain, modulated gain; Pol. Spec., polarization spectroscopy; AOTR, all-optical triple resonance; DR, double resonance; PFAOTR, perturbation-facilitated all-optical triple resonance; FT, Fourier transform; PFOODR, perturbation-facilitated optical-optical double resonance.

Fitted parameters for the potential function, for the and states. and are given in the following table. . The are in , while is dimensionless. Numbers in square brackets denote the power of 10. For the state, , the form applies, with values given below.

Fitted parameters for the potential function, for the and states. and are given in the following table. . The are in , while is dimensionless. Numbers in square brackets denote the power of 10. For the state, , the form applies, with values given below.

Potential parameters. and are in , in Å, and the parameters are in units . The parameters used in the exchange potential are dimensionless. work. Numbers in square brackets after a series of digits denote the power of 10, while parentheses denote uncertainty limits, which for the present work are discussed in the text.

Potential parameters. and are in , in Å, and the parameters are in units . The parameters used in the exchange potential are dimensionless. work. Numbers in square brackets after a series of digits denote the power of 10, while parentheses denote uncertainty limits, which for the present work are discussed in the text.

Deperturbed values, from previous studies and this one, perturbation shifts, values, and turning points for the state. eigenvalue for minus the deperturbed value, . The last two columns give the inner and outer turning points at the energies given in column 5. All quantities are in except for which are in Å.

Deperturbed values, from previous studies and this one, perturbation shifts, values, and turning points for the state. eigenvalue for minus the deperturbed value, . The last two columns give the inner and outer turning points at the energies given in column 5. All quantities are in except for which are in Å.

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