Total Energy Expression and Chemical Bonding in Hydrogen from The Prince Equation

We derive the total energy expression for hydrogen by properly maintaining the asymmetric scalar-vector nature of The Prince Equation. The requirement to treat radial (scalar from dot product) and tangential (vector from cross product) components differently leads to a total energy expression δE(T) = −hv(r) cosθ/[r(1 - sin θ)] with θ = n/(2π). This expression naturally creates confinement at n(max) = 8, explains hydrogen’s chemical reactivity through the n = 8 to 9 transition zone, and unifies covalent, ionic, and van der Waals bonding as different manifestations of the geometric transition from bound to free states.