energygradient

Energy gradient is the spatial rate of change of the energy of a system with respect to its coordinates. If the energy is given as E(r) where r is a position vector, the energy gradient is the vector ∇E(r) whose components are partial derivatives. It points in the direction of the greatest increase of energy; the opposite direction, −∇E(r), is the direction of steepest descent.

In conservative mechanical systems, the force acting on a particle due to a potential energy U is F = −∇U. Thus, the gradient of potential energy determines the motion: particles tend to move toward lower energy regions along the gradient flow.

Applications include the study of potential energy surfaces in chemistry and physics, where stationary points are found by solving ∇E = 0. In computational chemistry and physics, the energy gradient is used in geometry optimization, molecular dynamics, and transition-state searches. The magnitude of the gradient indicates how far a system is from equilibrium.

Gradient information can be obtained analytically from the energy function, or numerically by finite differences. In modern simulations, automatic differentiation or analytical gradients are preferred for accuracy and performance.

Related concepts include energy landscapes, gradient descent methods, and gradient flows in non-equilibrium systems; the energy gradient is a fundamental quantity linking energy to motions and stability.