Figure compares 𝑓𝑓(𝑛𝑛) of the original method and method 3. Since 𝑓𝑓(𝑛𝑛) ≤ 𝑛𝑛 for 𝑓𝑓(𝑛𝑛) under this definition, contribution of 𝐷𝐷(𝐫𝐫) from each natural orbital will always be less or equal to that of 𝜌𝜌(𝐫𝐫). This makes 𝐷𝐷(𝐫𝐫) ≤ 𝜌𝜌(𝐫𝐫) for all points in space. However, this definition of 𝑓𝑓(𝑛𝑛) gives discontinuities at points where 𝑛𝑛 = 1. Generally, the potential energy surface and wave function coefficients are smooth functions (i.e. functions with continuous first derivative) of the nuclear coordinates, except at conical intersections. Because the natural orbital occupation number is a function of the nuclear coordinates, by applying the chain rule, one can show that a non-smooth 𝑓𝑓(𝑛𝑛) can result in non-smooth wave function coefficients, and further result in nonsmooth potential energy surface. Because non-smooth 𝑓𝑓(𝑛𝑛) may appear in regions that are not conical intersections, method 3 may lead to unphysical non-smooth potential energy surfaces. An example of where the f(n) of method 3 leads to a non-smooth potential energy surface is any trajectory path along which one of the natural orbital occupation number passes through 1. Method 4 modifies the effective number of unpaired electrons such that we obtain a smooth DC functional.