The photoluminescence (PL) spectrum of Cu-containing silicon has a sharp zero-phonon (ZP) band at 1.014 eV, whose center called CuPL has the local symmetry C
3v
. Recent studies of experiment and theory revealed that the CuPL center is attributed to the Cu(s)Cu3(i) complex, which is composed of three interstitial Cu(i) atoms around a substitutional Cu(s) atom. This complex (called C-type) has the desired symmetry. However, in this study, we show that the lowest-energy structure is different. The tetrahedral structure Cu4, called T-type, has the lowest energy, with the value being 0.26 eV lower than that of C-type. Between these two types, there is an energy barrier of 0.14 eV, which allows C-type to exist in a metastable state. Details of the electronic properties of the T-type complex are reported, by comparing with C-type and other isovalent complexes. Whereas the Cu4 tetrahedron is incorporated in silicon in a manner compatible with the tetrahedral network, it also has its own molecular orbitals that exhibit metallic characteristics, in contrast to other complexes. The ZP of the PL spectrum is likely ascribed to the backflow mode of the Cu4 tetrahedron.