VEff

VEff, short for the effective potential (often written V_eff), is a concept used across physics to encapsulate multiple interactions into a single potential function that governs motion or the behavior of a system in a simplified form. It serves as a convenient tool for analyzing complex dynamics by lumping together centrifugal, mean-field, and exchange–correlation effects.

In classical mechanics, V_eff arises in central-force problems. For a particle of mass m moving in a central potential V(r) with angular momentum L, the radial motion is described by the effective potential V_eff(r) = V(r) + L^2/(2 m r^2). The term L^2/(2 m r^2) represents the centrifugal barrier. V_eff determines turning points, radial stability of orbits, and the shape of trajectories.

In quantum mechanics and atomic physics, V_eff is used in the radial part of the Schrödinger equation for particles in central fields. The angular momentum barrier modifies the potential felt by the radial wavefunction, influencing bound states and scattering. In computational methods such as density functional theory, V_eff denotes the Kohn–Sham effective potential: V_eff[n](r) = V_ext(r) + V_H[n](r) + V_xc[n](r), where V_ext is the external potential, V_H is the Hartree (electrostatic) term, and V_xc is the exchange–correlation potential. This V_eff is the input to the noninteracting reference system whose orbitals reproduce the true electron density n(r) in a self-consistent cycle.

VEff is generally not directly observable; it depends on the model and the chosen approximations. It is, however, a central concept for simplifying and solving problems in dynamics and electronic structure.

See also: effective potential energy, Kohn–Sham potential, Hartree potential, exchange–correlation potential.