Theoretical calculations of the Landé <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>g</mml:mi> </mml:math> -factors, electric quadrupole moments, and quadratic Zeeman shift coefficients of the clock states in the <mml:math xmlns:m

In this study, the multiconfiguration Dirac-Hartree-Fock method was employed to calculate the Landé <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mi>g</a:mi> </a:math> -factors and electric-quadrupole moments of the <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mrow> <b:mn>5</b:mn> <b:mi>d</b:mi> <b:mn>6</b:mn> <b:mi>s</b:mi> <b:msup> <b:mspace width="4pt"/> <b:mn>3</b:mn> </b:msup> <b:msub> <b:mi>D</b:mi> <b:mrow> <b:mn>1</b:mn> <b:mo>,</b:mo> <b:mn>2</b:mn> <b:mo>,</b:mo> <b:mn>3</b:mn> </b:mrow> </b:msub> </b:mrow> </b:math> and <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mrow> <d:msup> <d:mrow/> <d:mn>1</d:mn> </d:msup> <d:msub> <d:mi>D</d:mi> <d:mn>2</d:mn> </d:msub> </d:mrow> </d:math> states, as well as the quadratic Zeeman shift coefficients of the <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:mrow> <e:msup> <e:mrow/> <e:mn>3</e:mn> </e:msup> <e:msub> <e:mi>D</e:mi> <e:mrow> <e:mn>1</e:mn> <e:mo>,</e:mo> <e:mn>2</e:mn> </e:mrow> </e:msub> </e:mrow> </e:math> clock states in <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"> <f:msup> <f:mrow> <f:mi>Lu</f:mi> </f:mrow> <f:mo>+</f:mo> </f:msup> </f:math> . The electron correlation, particularly higher-order electron correlation effects, along with the Breit interaction and quantum electrodynamics corrections, are systematically taken into account. Our calculated Landé <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"> <g:mi>g</g:mi> </g:math> -factors, electric-quadrupole moments, and quadratic Zeeman shift coefficients for the <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"> <h:mrow> <h:mn>5</h:mn> <h:mi>d</h:mi> <h:mn>6</h:mn> <h:mi>s</h:mi> <h:msup> <h:mspace width="4pt"/> <h:mn>3</h:mn> </h:msup> <h:msub> <h:mi>D</h:mi> <h:mrow> <h:mn>1</h:mn> <h:mo>,</h:mo> <h:mn>2</h:mn> </h:mrow> </h:msub> </h:mrow> </h:math> clock states are in agreement with experimental measurements within <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"> <j:mrow> <j:mn>1</j:mn> <j:mo>%</j:mo> </j:mrow> </j:math> . The influence of hyperfine interactions and perturbing states on these parameters was also analyzed. Our results demonstrate that both the magnetic-dipole and the electric-quadrupole hyperfine interactions contribute significantly to the hyperfine-induced Landé <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"> <k:mi>g</k:mi> </k:math> -factors and electric-quadrupole moments of the <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"> <l:mrow> <l:msup> <l:mrow/> <l:mn>3</l:mn> </l:msup> <l:msub> <l:mi>D</l:mi> <l:mrow> <l:mn>1</l:mn> <l:mo>,</l:mo> <l:mn>2</l:mn> </l:mrow> </l:msub> </l:mrow> </l:math> states. In particular, for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mmultiscripts> <m:mi>Lu</m:mi> <m:none/> <m:mo>+</m:mo> <m:mprescripts/> <m:none/> <m:mn>176</m:mn> </m:mmultiscripts> </m:math> , the electric-quadrupole hyperfine interaction accounts for approximately <n:math xmlns:n="http://www.w3.org/1998/Math/MathML"> <n:mrow> <n:mn>15</n:mn> <n:mo>%</n:mo> </n:mrow> </n:math> of the total hyperfine-induced electric-quadrupole moment of the <o:math xmlns:o="http://www.w3.org/1998/Math/MathML"> <o:mrow> <o:msup> <o:mrow/> <o:mn>3</o:mn> </o:msup> <o:msub> <o:mi>D</o:mi> <o:mrow> <o:mn>1</o:mn> <o:mo>,</o:mo> <o:mn>2</o:mn> </o:mrow> </o:msub> </o:mrow> </o:math> states. This study provides valuable theoretical support for evaluating frequency shifts in the <p:math xmlns:p="http://www.w3.org/1998/Math/MathML"> <p:msup> <p:mrow> <p:mi>Lu</p:mi> </p:mrow> <p:mo>+</p:mo> </p:msup> </p:math> optical clock.