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I know from de Gennes' book [1] that the (magnetic) susceptibility tensor $\chi_{\alpha\beta}$ of an uniaxial liquid crystal in its nematic phase can be written as \begin{align} \chi_{\alpha\beta}&=\left(\frac{\chi_{xx}+\chi_{yy}+\chi_{zz}}{3}\right)\delta_{\alpha\beta}+\left(\chi_\|-\chi_\bot\right)\left(n_\alpha n_\beta-\frac{\delta_{\alpha\beta}}{3}\right)\\ &=A\delta_{\alpha\beta}+BQ_{\alpha\beta}, \end{align} where $Q_{\alpha\beta}=\left(n_\alpha n_\beta-\frac{\delta_{\alpha\beta}}{3}\right)$ is the tensorial order parameter, i.e., the anti-symmetric part of $\chi_{\alpha\beta}$ and $\chi_\|$ and $\chi_\bot$ are the molecular susceptibilities parallel and perpendicular to the director $\hat{n}$.

\begin{align}\end{align} Thus, what is the equivalent of this susceptibility for a biaxial nematic liquid crystal? As a trying, what is $B$ if I replace $Q_{\alpha\beta}$ by the ordering matrix [1] $S^{\alpha\beta}_{ij}=\frac{1}{2}\left(3i_\alpha j_\beta-\delta_{\alpha\beta}\delta_{ij}\right)$, regarding that $\vec{a}, \vec{b}, \vec{c}$ are the three orthogonal unit vectors linked to the molecule, with $i,j=a,b,c$ and $\alpha,\beta=x,y,z$? Can I expect that $B$ is a combination of the molecular susceptibilities $\chi_a, \chi_b, \chi_c$ on the $\vec{a}, \vec{b}, \vec{c}$ directions? If so, which combination is it?

UPDATE 2. I am more and more enjoying the possibility of using the ordering matrix $S^{\alpha\beta}_{ ij}$.

UPDATE 1. In [2] it is said that the diamagnetic anisotropy is written as $\Delta\chi=\chi_{zz}+\frac{1}{2}\left(\chi_{xx}+\chi_{yy}\right)$. If $\vec{a}, \vec{b}, \vec{c} $ are aligned to $\hat{x}, \hat{y}, \hat{z}$, can I regard $B\equiv\Delta\chi=\chi_{zz}+\frac{1}{2}\left(\chi_{xx}+\chi_{yy}\right)$ as the combination I'm looking for?

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The answer is it depends on the molecule. If it is parallelogram, one can set three director fields $\mathbf{n}$, $\mathbf{l}$ and $\mathbf{m}$ and a tensor order parameter $Q_{ij}=(-1/3)(Q-P) l_{i} l_j+(-1/3)(Q+P)m_im_j+(2/3)Qn_in_j$, as in the equation (B.I.9) of Oswald, Patrick, and Pawel Pieranski. Liquid Crystals: Concepts and Physical Properties Illustrated by Experiments, Two Volume Set. Crc Press, 2018.

For bent-core molecules, one can use PhysRevE.66.031704 - Theory of bent-core liquid-crystal phases and phase transitions

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