For isolated molecules at zero temperature, this function is ill-defined because of the derivative discontinuity. To resolve this difficulty, directional derivatives are defined by approaching from the left-handed, 𝑓 -(𝐫), and right-hand, 𝑓 + (𝐫), sides. Fukui function 𝑓 + (𝐫) measures the response of the electron density change following an addition of an electron, i.e., the propensity of nucleophilic (electron-gain) attack, whereas 𝑓 -(𝐫) is a measure of the electrophilic attack. In the finite difference approximation of a system with N electrons, these directional derivatives can be expressed by the electron density with N+1, N, and N-1 electrons, N+1(r), N(r), N-1(r), but in the frozen-orbital approximation, they can be obtained by the frontier orbitals density, 𝜌 𝐻𝑂𝑀𝑂 (𝐫) and 𝜌 𝐿𝑈𝑀𝑂 (𝐫),