Abstract
A theoretical investigation of the electronic structure of the NaK molecule including spin-orbit effects has been performed for the 34 Ω^{(+/−)} states dissociating adiabatically into the limits up to Na(3s ^{2}S_{1/2}) + K(3d ^{2}D_{3/2}) from both an ab initio approach and a long-range model. Equilibrium distances, transition energies, harmonic frequencies as well as depths of wells and heights of humps are reported for all the states. Formulas for calculating the long-range energies for all the 0^{+/−}, 1, 2, and 3 states under investigation are also displayed. They are expressed in terms of the C_{n} (n = 6,8, …) long-range coefficients and exchange integrals for the ^{2S+1}Λ^{(+)} parent states, available from literature. As present data could help experimentalists we make available extensive tables of energy values versus internuclear distances in our database at the web address: http://www-lasim.univ-lyon1.fr/spip.php?rubrique99.
The authors thank the “Pôle Scientifique de Modélisation Numérique (PSMN)” at Lyon, France, for generous computational facilities.
I. INTRODUCTION
II. THEORETICAL APPROACH
A. Quantum chemistry calculations
B. Long-range calculations
III. RESULTS
A. Ab initio results
B. Long-range model results
IV. CONCLUSION
Key Topics
- Ab initio calculations
- 20.0
- Spin orbit interactions
- 13.0
- Dissociation
- 8.0
- Dissociation energies
- 6.0
- Databases
- 4.0
Figures
Potential energy curves for the ^{1,3}Λ^{(+)}/Ω^{(+/−)} states dissociating adiabatically to Na(3s ^{2}S_{1/2}) + K(4p ^{2}P_{J}). (a) In the range 2–10 Å; the Ω^{(+/−)} components (marked by points) are quoted in brackets for each ^{1,3}Λ^{(+)} state and for various ranges of R; points ● for Ω = 0^{+}, points ▲ for Ω = 0^{−}, points ○ for Ω = 1, and points × for Ω = 2. (b) In the range 8–20 Å; long-dashed lines for Ω = 0^{+}, short-dashed lines for Ω = 0^{−}, full lines for Ω = 1, dotted line for Ω = 2. The ^{1,3}Λ^{(+)} states are indicated by points; points □ for (2)^{1}Σ^{+}, points ■ for (2)^{3}Σ^{+}, points * for (1)^{1}Π, and points × for (1)^{3}Π.
Potential energy curves for the ^{1,3}Λ^{(+)}/Ω^{(+/−)} states dissociating adiabatically to Na(3s ^{2}S_{1/2}) + K(4p ^{2}P_{J}). (a) In the range 2–10 Å; the Ω^{(+/−)} components (marked by points) are quoted in brackets for each ^{1,3}Λ^{(+)} state and for various ranges of R; points ● for Ω = 0^{+}, points ▲ for Ω = 0^{−}, points ○ for Ω = 1, and points × for Ω = 2. (b) In the range 8–20 Å; long-dashed lines for Ω = 0^{+}, short-dashed lines for Ω = 0^{−}, full lines for Ω = 1, dotted line for Ω = 2. The ^{1,3}Λ^{(+)} states are indicated by points; points □ for (2)^{1}Σ^{+}, points ■ for (2)^{3}Σ^{+}, points * for (1)^{1}Π, and points × for (1)^{3}Π.
Potential energy curves for the ^{1,3}Λ^{(+)}/Ω^{(+/−)} states dissociating adiabatically to Na(3p ^{2}P_{J}) + K(4s ^{2}S_{1/2}) and for the (4)^{1}Σ^{+} state dissociating adiabatically to Na(3s ^{2}S_{1/2}) + K(5s^{2}S_{1/2}). (a) In the range 4–13 Å; the Ω^{(+/−)} components (marked by points) are quoted in brackets for each ^{1,3}Λ^{(+)} state and for various ranges of R; points ● for Ω = 0^{+}, points × for Ω = 0^{−}, points ○ for Ω = 1, and points Δ for Ω = 2. The range 9–13 Å is enlarged in the insert, showing the crossings of the (2)^{3}Π with the (3)^{3}Σ^{+} and the (3)^{1}Σ^{+} PECs. (b) In the range 8–18 Å; long-dashed lines for Ω = 0^{+}, short-dashed lines for Ω = 0^{−}, full lines for Ω = 1, dotted line for Ω = 2. The ^{1,3}Λ^{(+)} states are indicated by points; points □ for (3)^{1}Σ^{+}, points ■ for (3)^{3}Σ^{+}, points * for (2)^{1}Π, and points × for (2)^{3}Π.
Potential energy curves for the ^{1,3}Λ^{(+)}/Ω^{(+/−)} states dissociating adiabatically to Na(3p ^{2}P_{J}) + K(4s ^{2}S_{1/2}) and for the (4)^{1}Σ^{+} state dissociating adiabatically to Na(3s ^{2}S_{1/2}) + K(5s^{2}S_{1/2}). (a) In the range 4–13 Å; the Ω^{(+/−)} components (marked by points) are quoted in brackets for each ^{1,3}Λ^{(+)} state and for various ranges of R; points ● for Ω = 0^{+}, points × for Ω = 0^{−}, points ○ for Ω = 1, and points Δ for Ω = 2. The range 9–13 Å is enlarged in the insert, showing the crossings of the (2)^{3}Π with the (3)^{3}Σ^{+} and the (3)^{1}Σ^{+} PECs. (b) In the range 8–18 Å; long-dashed lines for Ω = 0^{+}, short-dashed lines for Ω = 0^{−}, full lines for Ω = 1, dotted line for Ω = 2. The ^{1,3}Λ^{(+)} states are indicated by points; points □ for (3)^{1}Σ^{+}, points ■ for (3)^{3}Σ^{+}, points * for (2)^{1}Π, and points × for (2)^{3}Π.
Energy differences E_{ ab initio } –E_{LR} for the Ω states corresponding to the dissociation limit K(4s^{2}S_{1/2}) + Na(3p ^{2}P_{J}), for E_{LR} evaluated with literature C_{n} (lines) or with fitted C_{n} (points). Full lines and × points for the three Ω = 1 states, dashed-dotted lines and ■ points for the two Ω = 0^{+} states, dashed lines and o points for the two Ω = 0^{−} states and dotted line and ● points for the Ω = 2 state. The position of the Le Roy limit is indicated by a vertical line.
Energy differences E_{ ab initio } –E_{LR} for the Ω states corresponding to the dissociation limit K(4s^{2}S_{1/2}) + Na(3p ^{2}P_{J}), for E_{LR} evaluated with literature C_{n} (lines) or with fitted C_{n} (points). Full lines and × points for the three Ω = 1 states, dashed-dotted lines and ■ points for the two Ω = 0^{+} states, dashed lines and o points for the two Ω = 0^{−} states and dotted line and ● points for the Ω = 2 state. The position of the Le Roy limit is indicated by a vertical line.
Tables
The ^{2S+1}Λ^{(+)} and Ω^{(+/−)} molecular states correlated adiabatically to the dissociation limits up to K(3d ^{2}D_{3/2}) + Na(3s ^{2}S_{1/2}). Experimental dissociation energies ΔE measured from the lowest limit are quoted, together with that for the next asymptote.
The ^{2S+1}Λ^{(+)} and Ω^{(+/−)} molecular states correlated adiabatically to the dissociation limits up to K(3d ^{2}D_{3/2}) + Na(3s ^{2}S_{1/2}). Experimental dissociation energies ΔE measured from the lowest limit are quoted, together with that for the next asymptote.
Transition energies T_{e} (in cm^{−1}) evaluated from the bottom of the ground state (1)0^{+}, equilibrium distances R_{e} (in Å), harmonic frequencies ω _{e} (in cm^{−1}), and differences D_{e} (in cm^{−1}) between the energy at the adiabatic dissociation limit and the energy at the position of the identified structure (well or hump) for the Ω^{(+/−)} states. Dissociation limits are quoted through K(nl ^{2}L_{J}) + Na(n′l ′ ^{2}L′_{J′}).^{ 2S+1}Λ^{(+)} main parents as well as crossings between ^{2S+1}Λ^{(+)} states, identified at the position of the structure (well or hump) are quoted in the last column.
Transition energies T_{e} (in cm^{−1}) evaluated from the bottom of the ground state (1)0^{+}, equilibrium distances R_{e} (in Å), harmonic frequencies ω _{e} (in cm^{−1}), and differences D_{e} (in cm^{−1}) between the energy at the adiabatic dissociation limit and the energy at the position of the identified structure (well or hump) for the Ω^{(+/−)} states. Dissociation limits are quoted through K(nl ^{2}L_{J}) + Na(n′l ′ ^{2}L′_{J′}).^{ 2S+1}Λ^{(+)} main parents as well as crossings between ^{2S+1}Λ^{(+)} states, identified at the position of the structure (well or hump) are quoted in the last column.
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