| Summary: | 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.
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