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From the first-principles calculations, we have studied the electronic andmagnetic structures of compound Na$_2$Ti$_2$Pn$_2$O (Pn = As or Sb). We findthat in the ground state Na$_2$Ti$_2$As$_2$O is a blocked checkerboardantiferromagnetic semiconductor with a small band gap of about 0.15 eV, incontrast, Na$_2$Ti$_2$Sb$_2$O is a bi-collinear antiferromagnetic semimetal,both with a small moment of about 0.5$\mu_B$ around each Ti atom. We show thatthere is a strong Fermi surface nesting in Na$_2$Ti$_2$Pn$_2$O. And we verifythat the blocked checkerboard and bi-collinear antiferromagnetic states bothare the spin density waves induced by the Fermi surface nesting. A tetramerstructural distortion is found in company with the formation of a blockedcheckerboard antiferromagnetic order, in good agreement with the experimentallyobserved commensurate structural distortion but with space group symmetryretained after the anomaly happening. Further analysis and discussion inconnection with experimental observations are given as well.

Layered pnictide-oxide Na$_2$Ti$_2$Pn$_2$O (Pn=As, Sb): a paradigm for spin density waves