Many-electron formulation of the method of interatomic tunneling currents introduced in our earlier work [J. Chem. Phys. 104, 8424 (1996); 105, 10819 (1996)] for the description of long-range electron tunneling in large molecules such as proteins or DNA is proposed. The tunneling currents can be used both for calculation of the tunneling matrix element and for the description of the spatial distribution of tunneling pathways at the atomic level of resolution. It is shown that the tunneling currents can be expressed as a matrix element of a certain (current) operator evaluated between two diabatic nonorthogonal one- or multideterminant wave functions of the initial and final states of the electrons in the system. These states can be found in the standard ground state energy minimization calculations. Explicit expressions for the currents in terms of the atomic basis functions and the transformation matrices to molecular orbitals of the donor and acceptor states are given. Thus, the proposed theory provides a method that allows ordinary electronic structure calculations to be utilized for studies of tunneling dynamics in many-electron systems. All electron-electron interactions are included in the expressions for currents at the Hartree-Fock level, so that electron polarization effects arising due to interaction of the tunneling electron and other electrons in the system are taken into account in such a description.