Fine-Tuning of Atomic Energies in Relativistic Multiconfiguration Calculations
Ab initio calculations sometimes do not reproduce the experimentally observed energy separations at a high enough accuracy. Fine-tuning of diagonal elements of the Hamiltonian matrix is a process which seeks to ensure that calculated energy separations of the states that mix are in agreement with ex...
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MDPI AG
2023-04-01
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| author | Yanting Li Gediminas Gaigalas Wenxian Li Chongyang Chen Per Jönsson |
| author_facet | Yanting Li Gediminas Gaigalas Wenxian Li Chongyang Chen Per Jönsson |
| author_sort | Yanting Li |
| collection | DOAJ |
| description | Ab initio calculations sometimes do not reproduce the experimentally observed energy separations at a high enough accuracy. Fine-tuning of diagonal elements of the Hamiltonian matrix is a process which seeks to ensure that calculated energy separations of the states that mix are in agreement with experiment. The process gives more accurate measures of the mixing than can be obtained in ab initio calculations. Fine-tuning requires the Hamiltonian matrix to be diagonally dominant, which is generally not the case for calculations based on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>j</mi><mi>j</mi></mrow></semantics></math></inline-formula>-coupled configuration state functions. We show that this problem can be circumvented by a method that transforms the Hamiltonian in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>j</mi><mi>j</mi></mrow></semantics></math></inline-formula>-coupling to a Hamiltonian in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><mi>S</mi><mi>J</mi></mrow></semantics></math></inline-formula>-coupling for which fine-tuning applies. The fine-tuned matrix is then transformed back to a Hamiltonian in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>j</mi><mi>j</mi></mrow></semantics></math></inline-formula>-coupling. The implementation of the method into the General Relativistic Atomic Structure Package is described and test runs to validate the program operations are reported. The new method is applied to the computation of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><msup><mi>s</mi><mn>2</mn></msup><msup><mspace width="3.33333pt"></mspace><mn>1</mn></msup><msub><mi>S</mi><mn>0</mn></msub><mo>−</mo><mn>2</mn><mi>s</mi><mn>2</mn><mi>p</mi><msup><mspace width="3.33333pt"></mspace><mrow><mn>1</mn><mo>,</mo><mn>3</mn></mrow></msup><msub><mi>P</mi><mn>1</mn></msub></mrow></semantics></math></inline-formula> transitions in C III and to the computation of Rydberg transitions in B I, for which the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mi>s</mi><mn>2</mn><msup><mi>p</mi><mn>2</mn></msup><msup><mspace width="3.33333pt"></mspace><mn>2</mn></msup><msub><mi>S</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msub></mrow></semantics></math></inline-formula> perturber enters the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><msup><mi>s</mi><mn>2</mn></msup><mi>n</mi><mi>s</mi><msup><mspace width="3.33333pt"></mspace><mn>2</mn></msup><msub><mi>S</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msub></mrow></semantics></math></inline-formula> series. Improved convergence patterns and results are found compared with ab initio calculations. |
| first_indexed | 2024-03-11T05:15:22Z |
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| spelling | doaj.art-c0b5f22d1c6a444f9538d0c9f014adc82023-11-17T18:18:30ZengMDPI AGAtoms2218-20042023-04-011147010.3390/atoms11040070Fine-Tuning of Atomic Energies in Relativistic Multiconfiguration CalculationsYanting Li0Gediminas Gaigalas1Wenxian Li2Chongyang Chen3Per Jönsson4Shanghai EBIT Lab, Key Laboratory of Nuclear Physics and Ion-Beam Application, Institute of Modern Physics, Department of Nuclear Science and Technology, Fudan University, Shanghai 200433, ChinaInstitute of Theoretical Physics and Astronomy, Vilnius University, 010222 Vilnius, LithuaniaNational Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, ChinaShanghai EBIT Lab, Key Laboratory of Nuclear Physics and Ion-Beam Application, Institute of Modern Physics, Department of Nuclear Science and Technology, Fudan University, Shanghai 200433, ChinaDepartment of Materials Science and Applied Mathematics, Malmö University, 20506 Malmö, SwedenAb initio calculations sometimes do not reproduce the experimentally observed energy separations at a high enough accuracy. Fine-tuning of diagonal elements of the Hamiltonian matrix is a process which seeks to ensure that calculated energy separations of the states that mix are in agreement with experiment. The process gives more accurate measures of the mixing than can be obtained in ab initio calculations. Fine-tuning requires the Hamiltonian matrix to be diagonally dominant, which is generally not the case for calculations based on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>j</mi><mi>j</mi></mrow></semantics></math></inline-formula>-coupled configuration state functions. We show that this problem can be circumvented by a method that transforms the Hamiltonian in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>j</mi><mi>j</mi></mrow></semantics></math></inline-formula>-coupling to a Hamiltonian in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><mi>S</mi><mi>J</mi></mrow></semantics></math></inline-formula>-coupling for which fine-tuning applies. The fine-tuned matrix is then transformed back to a Hamiltonian in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>j</mi><mi>j</mi></mrow></semantics></math></inline-formula>-coupling. The implementation of the method into the General Relativistic Atomic Structure Package is described and test runs to validate the program operations are reported. The new method is applied to the computation of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><msup><mi>s</mi><mn>2</mn></msup><msup><mspace width="3.33333pt"></mspace><mn>1</mn></msup><msub><mi>S</mi><mn>0</mn></msub><mo>−</mo><mn>2</mn><mi>s</mi><mn>2</mn><mi>p</mi><msup><mspace width="3.33333pt"></mspace><mrow><mn>1</mn><mo>,</mo><mn>3</mn></mrow></msup><msub><mi>P</mi><mn>1</mn></msub></mrow></semantics></math></inline-formula> transitions in C III and to the computation of Rydberg transitions in B I, for which the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mi>s</mi><mn>2</mn><msup><mi>p</mi><mn>2</mn></msup><msup><mspace width="3.33333pt"></mspace><mn>2</mn></msup><msub><mi>S</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msub></mrow></semantics></math></inline-formula> perturber enters the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><msup><mi>s</mi><mn>2</mn></msup><mi>n</mi><mi>s</mi><msup><mspace width="3.33333pt"></mspace><mn>2</mn></msup><msub><mi>S</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msub></mrow></semantics></math></inline-formula> series. Improved convergence patterns and results are found compared with ab initio calculations.https://www.mdpi.com/2218-2004/11/4/70fine-tuningmulticonfiguration Dirac–Hartree–Fockjj-couplingLSJ-couplingcoupling transformation |
| spellingShingle | Yanting Li Gediminas Gaigalas Wenxian Li Chongyang Chen Per Jönsson Fine-Tuning of Atomic Energies in Relativistic Multiconfiguration Calculations Atoms fine-tuning multiconfiguration Dirac–Hartree–Fock jj-coupling LSJ-coupling coupling transformation |
| title | Fine-Tuning of Atomic Energies in Relativistic Multiconfiguration Calculations |
| title_full | Fine-Tuning of Atomic Energies in Relativistic Multiconfiguration Calculations |
| title_fullStr | Fine-Tuning of Atomic Energies in Relativistic Multiconfiguration Calculations |
| title_full_unstemmed | Fine-Tuning of Atomic Energies in Relativistic Multiconfiguration Calculations |
| title_short | Fine-Tuning of Atomic Energies in Relativistic Multiconfiguration Calculations |
| title_sort | fine tuning of atomic energies in relativistic multiconfiguration calculations |
| topic | fine-tuning multiconfiguration Dirac–Hartree–Fock jj-coupling LSJ-coupling coupling transformation |
| url | https://www.mdpi.com/2218-2004/11/4/70 |
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