Energy Minimization and Geometry Optimization
The basic task in the computational portion of MM is to minimize the strain energy of the molecule by altering the atomic positions to optimal geometry. This means minimizing the total nonlinear strain energy represented by the FF equation with respect to the independent variables, which are the Cartesian coordinates of the atoms (Altona and Faber, 1974). The following issues are related to the energy minimization of a molecular structure:
• The most stable configuration of a molecule can be found by minimizing its free energy, G.
• Typically, the energy E is minimized by assuming the entropy effect can be neglected.
• At a minimum of the potential energy surface, the net force on each atom vanishes, therefor the stable configuration.
• Because the energy zero is arbitrary, the calculated energy is relative. It is meaningful only to compare energies calculated for different configurations of chemically identical systems.
• It is difficult to determine if a particular minimum is the global minimum, which is the lowest energy point where force is zero and second derivative matrix is positive definite. Local minimum results from the net zero forces and positive definite second derivative matrix, and saddle point results from the net zero forces and at least one negative eigenvalue of the second derivative matrix.
The most widely used methods fall into two general categories: (1) steepest descent and related methods such as conjugate gradient, which use first derivatives, and (2) Newton - Raphson procedures, which additionally use second derivatives.
The steepest descent method (Wiberg, 1965) depends on (1) either calculating or estimating the first derivative of the strain energy with respect to each coordinate of each atom and (2) moving the atoms. The derivative is estimated for each coordinate of each atom by incrementally moving the atom and storing the resultant strain energy change. The atom is then returned to its original position, and the same calculation is repeated for the next atom. After all the atoms have been tested, their positions are all changed by a distance proportional to the derivative calculated in step 1. The entire cycle is then repeated. The calculation is terminated when the energy is reduced to an acceptable level. The main problem with the steepest descent method is that of determining the appropriate step size for atom movement during the derivative estimation steps and the atom movement steps. The sizes of these increments determine the efficiency of minimization and the quality of the result. An advantage of the first-derivative methods is the relative ease with which the force field can be changed.
The conjugate gradient method is a first-order minimization technique. It uses both the current gradient and the previous search direction to drive the minimization. Because the conjugated gradient method uses the minimization history to calculate the search direction and contains a scaling factor for determining step size, the method converges faster and makes the step sizes optimal as compared to the steepest descent technique. However, the number of computing cycles required for a conjugated gradient calculation is approximately proportional to the number of atoms (N), and the time per cycle is proportional to N2. The Fletcher-Reeves approach chooses a descent direction to lower energy by considering the current gradient, its conjugate, and the gradient for the previous step. The Polak-Ribiere algorithm improves on the Fletcher-Reeves approach by additional consideration of the previous conjugate and tends to converge more quickly.
The Newton - Raphson methods of energy minimization (Berkert and Allinger, 1982) utilize the curvature of the strain energy surface to locate minima. The computations are considerably more complex than the first-derivative methods, but they utilize the available information more fully and therefore converge more quickly. These methods involve setting up a system of simultaneous equations of size (3N — 6) (3N — 6) and solving for the atomic positions that are the solution of the system. Large matrices must be inverted as part of this approach.
The general strategy is to use steepest descents for the first 10-100 steps (500-1000 steps for proteins or nucleic acids) and then use conjugate gradients or Newton-Raphson to complete minimization for convergence (using RMS gradient or/and energy difference as an indicator). For most calculations, RMS gradient is set to 0.10 (you can use values greater than 0.10 for quick, approximate calculations). The calculated minimum represents the potential energy closest to the starting structure of a molecule. The energy minimization is often used to generate a structure at a stationary point for a subsequent single-point calculation or to remove excessive strain in a molecule, preparing it for a molecular dynamic simulation.
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