Current interest in the fabrication of organic nanostructures on silicon surface is focused on the self-directed growth of 1D molecular lines with predefined position, structure, composition, and the length on the H-terminated Si(100)-(2 x 1) surface. To date, no studies have succeeded in growing the molecular line across the dimer rows on Si(100)-(2 x 1)-H, which is highly desirable. Using scanning tunneling microscopy (STM), we studied a new molecular system (allyl mercaptan, CH2=CH-CH2-SH) that undergoes chain reaction across the dimer row on the Si(100)-(2 x 1)-H surface at 300 K and produces covalently bonded 1D molecular lines. In combination with the previous findings of chain reaction along the rows, the present observations of self-directed growth of 1D molecular lines across the dimer rows on the Si(100)-(2 x 1)-H surface provide a means to connect any two points (through molecular lines) on a 2D surface.