Quiz on Molecular Energy Minimization and Optimization Algorithms, Slides of Engineering Chemistry

Download Quiz on Molecular Energy Minimization and Optimization Algorithms and more Slides Engineering Chemistry in PDF only on Docsity! 1 QUIZ l Should MM energies be the same or different for the structure pairs shown? CH3 OH CH3 OH HO CH3 OH CH3 CH3 OH OH CH3 A B C Energy Minimization Potential Energy Surfaces Minima Saddle Points Maxima Energy Minimization Methods l Energy Minimization is used synonymously with geometry optimization l Derivative-based l Optimization algorithms that use derivatives of the energy function l Non derivative-based l Optimization algorithms that do not use derivatives of the energy function Simplex Algorithm l Simplex: A figure with one more interconnected vertex than the energy function has dimensions l A non-derivative method l Requires 4 vertices for Cartesian optimization l Three basic strategies l Reflection l Expansion l Contraction l Effective for bad geometries, very slow near minima Simplex Example 1D energy function = central torsion angle in butane Starting position = 135º Simplex needs 2 vertices, next point is 135 + xº (x is a small, often random number) Result of reflection Result of contraction docsity.com 2 Derivative Minimization Methods l First derivative l Indicates slope of energy surface = gradient l Gradient = 0 indicates maxima and saddle points as well as the minima we usually want l Second derivative l Differentiates between types of points with gradient = 0, indicates curvature l Positive curvature = minima l Negative curvature = maxima l Zero curvature = saddle points l Methods are assigned an order based on the highest derivative that they use Steepest Descent l A first-order method l Direction of net force is followed with: l An arbitrary step size l Line search Line Search q Requires three points that bracket the minima (second point must have lowest energy) q Two strategies n Iteratively select points between two lower-energy points (lots of function evaluations necessary n Fit a curve to the three points and use its minima as the next point selected Limitations of Steepest Descent l Can’t differentiate between maxima, minima and saddle points l Very slow at low gradient values (near minima) l Very inefficient for long, narrow energy wells Other Minimization Algorithms l Conjugate Gradient l Also a first-order method (will have same problem as steepest descent with maxima/minima/saddle points) l Uses gradients from two successive points to determine direction after first step – behaves in a less oscillatory fashion Other Minimization Algorithms II l Newton-Raphson, a second order method l Suitable for relatively small systems (~100 atoms) due the the way the derivatives are handled l Truncated Newton l Solves for the second derivative iteratively (truncated after some number of iterations) l Method of choice except for highly strained systems docsity.com