\frac{g^2_1 g^2_2 f_{\tilde{B}} f_{\tilde{H}} \,m^4_N}{4\pi m^2_W \cos^2 \beta \, m^4_H} \bigg(f_{T_s}+\frac{2}{27}f_{TG}\bigg)^2, \,\,\,\, (m_{\tilde{q}}\, \rm{large}, \cos \alpha \approx 1),
\label{case1}$$ where $f_{\tilde{B}}$ and $f_{\tilde{H}}$ denote the bino and higgsino fractions of the lightest neutralino and $\tan \beta$ is the ratio of the vevs of the two Higgs doublets in the MSSM. Note that the coupling involves the product of $f_{\tilde{B}}$ and $f_{\tilde{H}}$. Neutralinos that are purely gaugino-like or purely higgsino-like have zero cross section with nuclei. The fundamental reason for this is that the relevant vertex is gaugino-higgsino-Higgs.

If the heavier of the two CP-even Higgs bosons is very heavy and/or $\tan \beta$ is small, scattering with up-type quarks through light Higgs exchange can dominate: $$\sigma_{\chi N} \sim \frac{g^2_1 g^2_2  f_{\tilde{B}} f_{\tilde{H}} \, m^4_N}{4\pi m^2_W \, m^4_h} \bigg(f_{T_u}+\frac{4}{27}f_{TG}\bigg)^2, \,\,\,\, (m_{\tilde{q}}, m_H\, \rm{large}, \cos \alpha \approx 1).
\label{case2}$$ If $\tan \beta$ and $m_H$ are large and the squarks somewhat light, elastic scattering can instead be dominated by squark exchange: $$\sigma_{\chi N} \sim \frac{g^2_1 g^2_2  f_{\tilde{B}} f_{\tilde{H}} \, m^4_N}{4\pi m^2_W \cos^2 \beta \, m^4_{\tilde{q}}} \bigg(f_{T_s}+\frac{2}{27}f_{TG}\bigg)^2, \,\,\,\, (\tilde{q}\,\, \rm{dominated}, \tan \beta \gg 1).
\label{case3}$$ From these expressions [@scatteraq], it is clear that the direct detection of dark matter alone will not be very capable 