Ab initio study has been used to explore the problem if the Peierls instability in hydrogen chain presents or not. With a Gaussian-type-orbital basis set and periodicity of hydrogen chain, we construct a complete set of orthogonal and normalized wave functions in view of the second-neighbor approximation. Starting from the restricted Hartree-Fock scheme, we diagnoalize the Fock matrix analytically and calculate the LCAO-MO-SCF solutions. The dependence of the total energy on dimerization is analyzed, it is shown that the ground state of metal hydrogen chain is unstable, and that obvious dimerization may occur in infinite hydrogen chain. The Per erls instability presents in this system nevertheless.