Fine-Tuning of Atomic Energies in Relativistic Multiconfiguration Calculations
Abstract
:1. Introduction
2. Multiconfiguration Methods
Expansions over Configuration State Functions in - and -Coupling
3. Hamiltonian Matrix in - and -Coupling
4. Transition Parameters in - and -Coupling
5. Transformation between Coupling Schemes
6. Fine-Tuning of Eigenvalues
7. Fine-Tuning of Eigenvalues in -Coupling
8. Fine-Tuning of Eigenvalues in -Coupling
- arrange the full CSF expansion so that the CSFs in the multireference (MR) come first.
- compute the coupling transformation matrix between the -coupled CSFs in the MR and the corresponding -coupled CSFs.
- perform a relativistic CI calculation for the full CSF expansion. Save the Hamiltonian on disk in sparse format.
- read and transform the Hamiltonian submatrix corresponding to the CSFs in the MR from -coupling () to -coupling ( according to
- allow the user to fine-tune the diagonal elements of to yield .
- transform the fine-tuned Hamiltonian matrix back to -coupling according to
- perform a relativistic CI calculation for the full CSF expansion based on the Hamiltonian for which the submatrix corresponding to the CSFs in the MR was modified.
9. Program Implementation
10. Test-Run
11. Applications
11.1. Transition Rates in C III
11.2. Lifetimes of the Rydberg States in B I
12. Heavy and Complex Systems
13. Summary and Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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No | Pos | J | Parity | Configuration | RCI | NIST | fine-tune | |
(cm^−1) | (cm^−1) | (cm^−1) | ||||||
1 | 1 | 0 | + | 2s(2)_1S | 0.00 | 0.00 | ||
2 | 1 | 0 | − | 2s_2S.2p_3P | 52463.43 | 52367.06 | −96 | |
3 | 1 | 1 | − | 2s_2S.2p_3P | 52486.91 | 52390.75 | −96 | |
4 | 1 | 2 | − | 2s_2S.2p_3P | 52543.19 | 52447.11 | −96 | |
5 | 2 | 1 | − | 2s_2S.2p_1P | 102530.94 | 102352.04 | −179 |
State | Ab Initio | Fine-Tuned | NIST | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
3 | 4 | 5 | 6 | 7 | 3 | 4 | 5 | 6 | 7 | ||
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
52,321.81 | 52,445.12 | 52,411.16 | 52,407.97 | 52,463.43 | 52,366.94 | 52,367.32 | 52,367.24 | 52,367.23 | 52,367.82 | 52,367.06 | |
52,344.14 | 52,468.08 | 52,434.34 | 52,431.32 | 52,486.91 | 52,390.57 | 52,391.08 | 52,390.92 | 52,390.89 | 52,391.30 | 52,390.75 | |
52,398.05 | 52,523.24 | 52,490.00 | 52,487.35 | 52,543.19 | 52,446.97 | 52,447.43 | 52,447.27 | 52,447.31 | 52,447.57 | 52,447.11 | |
104,112.00 | 102,962.02 | 102,613.81 | 102,517.46 | 102,530.94 | 102,395.09 | 102,366.93 | 102,358.33 | 102,356.05 | 102,356.24 | 102,352.04 | |
880.92 | 312.23 | 136.21 | 89.86 | 122.24 | 21.53 | 7.45 | 3.15 | 2.01 | 2.16 |
State | Ab Initio | Fine-Tuned | NIST |
---|---|---|---|
0 | 0 | 0 | |
14.51 | 15.287 | ||
39,800.91 | 40,038.65 | 40,039.6907 | |
48,520.93 | 48,611.8663 | ||
48,522.60 | 48,613.6486 | ||
54,707.61 | 55,012.88 | 55,010.2338 | |
57,669.26 | 57,786.4336 | ||
57,669.86 | 57,787.0683 | ||
59,860.42 | 60,146.56 | 60,146.414 | |
61,306.69 | 61,433.59 | ||
61,306.97 | 61,433.59 | ||
62,254.45 | 62,476.47 | 62,482.167 | |
63,129.71 | 63,263.24 | ||
63,129.87 | 63,263.24 | ||
65,102.64 | 63,464.94 | 63,560.638 | |
63,526.70 | 64,166.09 | 64,156.017 |
State | Ab Initio | Fine-Tuned | Experiment | MCHF |
---|---|---|---|---|
4.11 | 3.99 | 4.0 (2) a | 3.97 b | |
9.86 | 8.67 | 8.7 (4) a | 8.59 b | |
17.6 | 12.2 | 11.0 (6) b | 11.3 b | |
25.1 | 8.40 | 7.7 (4) b | 7.65 b | |
3.24 | 3.29 | 3.3 (2) b | 3.65 b | |
3.6 (3) c | ||||
23.6 | 11.1 | 8.3 (4) b | 8.01 b |
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Li, Y.; Gaigalas, G.; Li, W.; Chen, C.; Jönsson, P. Fine-Tuning of Atomic Energies in Relativistic Multiconfiguration Calculations. Atoms 2023, 11, 70. https://doi.org/10.3390/atoms11040070
Li Y, Gaigalas G, Li W, Chen C, Jönsson P. Fine-Tuning of Atomic Energies in Relativistic Multiconfiguration Calculations. Atoms. 2023; 11(4):70. https://doi.org/10.3390/atoms11040070
Chicago/Turabian StyleLi, Yanting, Gediminas Gaigalas, Wenxian Li, Chongyang Chen, and Per Jönsson. 2023. "Fine-Tuning of Atomic Energies in Relativistic Multiconfiguration Calculations" Atoms 11, no. 4: 70. https://doi.org/10.3390/atoms11040070
APA StyleLi, Y., Gaigalas, G., Li, W., Chen, C., & Jönsson, P. (2023). Fine-Tuning of Atomic Energies in Relativistic Multiconfiguration Calculations. Atoms, 11(4), 70. https://doi.org/10.3390/atoms11040070