where ε αβ γ is the Levi-Civita symbol, ϕ µ ϕ ν ϕ κ ϕ τ is an Coulomb electron repulsion integral (ERI), and [L γ µν ] J is an element of the J-th Cholesky vector in AO basis. N CH the dimension of the Cholesky basis, i.e. the number of the Cholesky vectors. In ref. , we showed that chosen the Cholesky δ threshold, the errors ∆ SOC affecting the integrals evaluated by eq. ( ) is always ∆ SOC ≤ 2δ . Then, the 2e-SOC integrals in MO basis can be expressed on the Cholesky basis by substituting eq. ( ) into eq. ( ):