The potential and force acting on one electron within a molecule (PAEM and FAEM) have been investigated and analyzed. The PAEM, defined as the interaction energy on one electron provided by all the nuclei and the remaining electrons in a molecule, can be precisely expressed and calculated by ab initio method and our in-house program. Although the analysis of the scalar function PAEM is similar to that of the molecular electron density in the Bader's AIM theory, the former is distinct from the latter mainly in three points: (a) The minus gradient of the PAEM is the force acting on one electron within a molecule (FAEM). (b) The bond center is defined in terms of the features of FAEM and PAEM between two bonded atoms, and it is a two-dimensional attractive center whereas a nucleus is a three-dimensional attractive source for electrons. We have calculated the physical quantities of one electron at the bond center, such as Dpb, the Hessian matrix, and its eigenvalues. Interestingly, it is found that the force constant and frequency of the electron interflow around the bond center are well correlated with those corresponding quantities for the nuclear vibration which relate to the bond strength, for some series of diatomic molecules. (c) The bond center locates at a different point from that of the critical point of the electron density in the Bader's AIM theory, which will lead to different partitioning of the molecular space into the atomic regions.