To study the inter-ring sigma/(pi/pi) covalent interactions between non-radical pi-systems, five structures of cyclodimers of benzene (C6H6)2 with all the real frequencies, i.e. o-p'-dibenzene (), the pentacyclic dimer (), p-p'-dibenzene (), syn-o,o'-dibenzene (), and hexaprismane (), are obtained at the MP2/6-311G(d,p) level. Five inter-ring bonding mode types forming the inter-ring multicenter multielectron sigma/(pi/pi) covalent bonds are represented: , ring-edge type between a butterfly-shaped ring and a planar ring (4-center 4-electron bond); , edge-edge and ring-ring types between two identical butterfly-shaped rings (8-center 8-electron bond); , ring-ring type between two identical butterfly-shaped rings (4-center 4-electron bond); , edge-edge type between two identical planar rings (4-center 4-electron bond); and , face-face type between two identical planar rings (12-center 12-electron bond). The order of the large inter-ring interaction energies at the MP2/6-311+G(3d, 2p)+BF level is -99.15 (with two inter-ring C-C bonds)> -98.57 (with four C-C bonds)>-85.76 (with two C-C bonds)>-61.35 (with two C-C bonds)>-60.40 kcal mol(-1) (with six C-C bonds). However, this does not show an obvious relationship between the interaction energy and the number of the inter-ring C-C bonds. The reason is that the number of decisive influencing factors of the inter-ring interaction energy is not one but five: the number of the favorable inter-ring C-C single bonds, the number of the unfavorable four-membered rings themselves, the participating number of the four-membered rings in unfavorable interaction among those rings, the number of the favorable non-planar melted six-membered ring, and the weak inter-ring pi/pi interaction (between two pi bonds in different rings).