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In this study we implemented a massively parallel version of the direct Hartree-Fock(HF) method in GAMESS-US, which is one of the oldest and most popular open-source quantum chemical software packages. HF is one of the basic ab initio methods in quantum chemistry. In direct (diskless) HF method the computational cost is dominated by a Fock matrix construction step which is usually of supercomputer scale. This step requires calculation of O(N4) six-dimensional integrals corresponding to the Coulomb repulsion of electrons (electron repulsion integrals, ERI), where N is a number of basis functions used to characterize the system. In this work we developed a hybrid MPI/OpenMP Fock matrix code and optimized the performance of ERI code as well. We obtained a good scalability of the current implementation on Xeon Phi cores, as well as with multiple Xeon Phi chips running in native mode (OpenMP+MPI parallelization).