Emergence of superconductivity in the iron-based superconductor of "LaFeAsO" is a consequence of the characteristic behavior of antiferromagnetic structure at different triplet critical points. Here, we studied x(2)(c) (highest concentration bound) = 0.5. A new second antiferromagnetic (AFM) structure was found that is developed for three different geometrical arrangements of O/H, resulting in three different antiferromagnetic structures at x(2)(c) = 0.5. Both crystal and magnetic orderings are a consequence of the stripe like ordering in the stripe configuration of "H" and "0". Moreover, in our calculation a strong correlation was obtained between the magnetic and geometrical structures, resulting in the non-uniform magnetic moments of similar to 1.30 mu beta (AFM1-1) and similar to 1.66 mu beta (AFM1-3). This phenomenon results in the Jahn-Teller distortion, reflected in the non-uniformity of spin-induced distribution, in the range of 1.07 mu beta close to "H" and 1.53 mu beta close to "O", in order to stabilize the ground state of the magneto-crystalline structure. This newly found second antiferromagnetic ordering behaves as the ground state with a lower cohesive energy, resulting in a lower magnetic moment. The effects are more pronounced in the displacement of atomic positions, reflecting in the repulsion of intra-layer La atoms from each other and the attraction of inter-layer Fe atoms toward each other, in direction to induce the anisotropy variable of eta approximate to 0.78. (C) 2016 Elsevier B.V. All rights reserved.