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In this paper we study the Hartree-Fock type system as follows:\begin{equation*} \left\{ \begin{array}{ll} -\Delta u+u+\lambda \phi_{u,v}u=\left\vert u\right\vert ^{p-2}u+\beta \left\vert v\right\vert^{\frac{p}{2}}\left\vert u\right\vert ^{\frac{p}{2}% -2}u & \text{ in}\mathbb{R}^{3}, \\ -\Delta v+v+\lambda \phi _{u,v}v=\left\vert v\right\vert^{p-2}v+\beta \left\vert u\right\vert ^{\frac{p}{2}}\left\vert v\right\vert^{\frac{p}{2}% -2}v & \text{ in }\mathbb{R}^{3},% \end{array}% \right.\end{equation*}% where $\phi_{u,v}(x)=\int_{\mathbb{R}^{3}}\frac{u^{2}(y)+v^{2}\left( y\right) }{|x-y|}dy,$the parameters $\lambda,\beta >0$ and $2<p<4$. Such system is viewed as anapproximation of the Coulomb system with two particles appeared in quantummechanics, taking into account the Pauli principle. Its characteristic featurelies on the presence of the double coupled terms. When $2<p<3,$ we establishthe existence and multiplicity of nontrivial radial solutions, includingvectorial ones, in the radial space $% H_{r}$ by describing the internalrelationship between the coupling constants $\lambda $ and $\beta.$ When$2<p<4,$ we study the existence of vectorial solutions in the non-radial space$H$ by developing a novel constraint method, together with some new analysistechniques. In particular, when $3\leq p<4,$ a vectorial ground state solutionis found in $% H$, which is innovative as it was not discussed at all in anyprevious results. Our study can be regarded as an entire supplement in d'Aveniaet al. [J. Differential Equations 335 (2022) 580--614].

Vectorial ground state solutions for a class of Hartree-Fock type systems with the double coupled feature