Electronic energy scales of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>CrX</mml:mi> <mml:mn>3</mml:mn> </mml:msub> </mml:math> ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>X</mml:mi>

Chromium trihalides <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:mi>Cr</a:mi> <a:msub> <a:mi>X</a:mi> <a:mn>3</a:mn> </a:msub> </a:mrow> </a:math> ( <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mrow> <b:mi>X</b:mi> <b:mo>=</b:mo> <b:mtext>Cl</b:mtext> </b:mrow> </b:math> , Br, and I) have recently become a focal point of research due to their intriguing low-temperature, layer-dependent magnetic properties that can be manipulated by external stimuli. This makes them essential candidates for spintronic applications. Their magnetic orders are often related to the electronic structure parameters, such as spin-orbit coupling (SOC), Hund’s coupling ( <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:msub> <c:mi>J</c:mi> <c:mi>H</c:mi> </c:msub> </c:math> ), <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mrow> <d:mi>p</d:mi> <d:mo>−</d:mo> <d:mi>d</d:mi> </d:mrow> </d:math> covalency, and interorbital Coulomb interactions. Accurately determining such parameters is paramount for understanding <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:mrow> <e:mi>Cr</e:mi> <e:msub> <e:mi>X</e:mi> <e:mn>3</e:mn> </e:msub> </e:mrow> </e:math> physics. We have used high-resolution resonant inelastic x-ray scattering spectroscopy to study <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"> <f:mrow> <f:mi>Cr</f:mi> <f:msub> <f:mi>X</f:mi> <f:mn>3</f:mn> </f:msub> </f:mrow> </f:math> across phase transition temperatures. Ligand field multiplet calculations were used to determine the electronic structure parameters by incorporating the crystal field interactions in a distorted octahedral orientation with <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"> <g:msub> <g:mi>C</g:mi> <g:mn>3</g:mn> </g:msub> </g:math> symmetry. These methods provide the most detailed description of <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"> <h:mrow> <h:mi>Cr</h:mi> <h:msub> <h:mi>X</h:mi> <h:mn>3</h:mn> </h:msub> </h:mrow> </h:math> magneto-optical and electronic (energetic) terms to date. The crystal field distortion parameters <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"> <i:mrow> <i:mi>D</i:mi> <i:mi>σ</i:mi> </i:mrow> </i:math> and <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"> <j:mrow> <j:mi>D</j:mi> <j:mi>τ</j:mi> </j:mrow> </j:math> were experimentally determined, and the energies of <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"> <k:mi>d</k:mi> </k:math> orbitals have been reported. The spectroscopic measurements reveal an energy separation between spin-allowed quartet states and spin-forbidden doublet states, which increases upon going from <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"> <l:msub> <l:mi>CrCl</l:mi> <l:mn>3</l:mn> </l:msub> </l:math> to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>CrI</m:mi> <m:mn>3</m:mn> </m:msub> </m:math> . The role of SOC, for Cr <n:math xmlns:n="http://www.w3.org/1998/Math/MathML"> <n:mn>2</n:mn> <n:mrow> <n:mi>p</n:mi> </n:mrow> </n:math> orbitals, in spin-flip excitations has been demonstrated. The determined <o:math xmlns:o="http://www.w3.org/1998/Math/MathML"> <o:mrow> <o:mn>10</o:mn> <o:mi>D</o:mi> <o:mi>q</o:mi> </o:mrow> </o:math> values are in good agreement with the spectrochemical series, and Racah <p:math xmlns:p="http://www.w3.org/1998/Math/MathML"> <p:mrow> <p:mi>B</p:mi> </p:mrow> </p:math> follows the nephelauxetic effect. Such precise measurements offer insights into the energy design of spintronic devices that utilize quantum state tuning within two-dimensional magnetic materials.