Energieminima

Energieminima, the German term for energy minima, describes points on a system’s potential energy surface where the energy is at a local minimum with respect to small displacements in coordinates. They occur in molecular conformations, crystal structures, and mechanical systems. An energy minimum is stable against small perturbations at zero temperature.

Global minimum is the absolute lowest energy among all configurations; local minima are lower than neighboring configurations but not necessarily the lowest overall. Many systems have multiple local minima separated by energy barriers; at finite temperature, transitions occur via thermal activation. In chemistry and physics, Energieminima correspond to stable or metastable states, such as preferred molecular shapes or crystal arrangements.

Mathematically, a minimum is a point x where the gradient ∇V(x) = 0 and the Hessian ∂^2V/∂x^2 at x is positive definite (all eigenvalues positive). If any eigenvalue is zero or negative, it is a saddle point or maximum rather than a minimum. In high-dimensional systems, numerical optimization is typically used to locate minima.

Common algorithms include steepest descent and conjugate gradient methods; Newton or quasi-Newton methods such as BFGS can be faster but require more computation. For large systems, limited-memory variants (L-BFGS) are popular. Global search strategies—basin-hopping, genetic algorithms, simulated annealing, and metadynamics—help escape local minima. Verification of a true minimum often involves vibrational analysis to ensure all frequencies are real and positive.

Applications cover chemistry (stable molecular conformations and reaction intermediates), materials science (stable crystal structures and defects), and physics (rugged energy landscapes in spin or mechanical systems). Limitations arise from rugged, high-dimensional landscapes, dependence on initial guesses, and the need to account for temperature and entropic effects beyond the potential energy minimum.