Lecture 13. Geometry Optimization References Computational chemistry: Introduction to the theory and applications of molecular and quantum mechanics, E. Lewars (2004) Chapter Molecular Modeling, A. R. Leach (2nd ed. 2001), Chapter 5 (pp ) Essentials of computational chemistry. Theories and Models, C. J. Cramer, (2nd Ed. Wiley, 2004)Chapter 2.4 Introduction to Computational Chemistry, F. Jensen (1999), Chapter 14

Geometry Optimization. Minimization Method Non-derivative minimization method - Simplex Derivative Minimization Method First derivative method - Steepest descent (popular in MM far from minimum) - Conjugate gradient (popular in MM near minimum) Second derivative method - Newton-Raphson - Quasi-Newton: BFGS, DFP, MS, etc. (popular in QM)

Potential Energy Curve (1-Dimensional) A B R Simplest form: Harmonic Oscillator E = E(R) Simplified

Optimized in one step, if quadratic, from 1 st and 2 nd derivatives of energy True only near equilibrium structure Otherwise, optimized in several steps Constant only when quadratic End up with the minimum closest to the input structure (local minimum): No guarantee for the global minimum Second derivative method: Newton-Raphson (1D, Diatom)

Second derivative method: Newton-Raphson/Quasi-Newton Position (coordinates): 3N-dim. vector 1 st derivative (gradient): 3N-dim. vector 2 nd derivative (hessian): 3N  3N matrix - inverted Takes only few steps to converge near minimum, but can be unstable far from minimum Used for small molecules near minimum (after the first steps of steepest descent) (updated at each step analytically or numerically) (easily calculated analytically) from 1 st and 2 nd derivatives of energy 1-Dim (diatom; N=2; #(degree of freedom)=3N-5=1) N-atom; #(degree of freedom)=3N or 3N-6 (non-linear) or 3N-5 (linear) (time- and memory-consuming for large systems)

Second derivative method: Newton-Raphson example from (9,9)

Second derivative method: Newton-Raphson Summary First derivative gives direction of vector. Second derivative gives curvature of the direction vector. This allows for the minimum to be guessed along the line searched. The minimum of a quadratic function can be reached in one step. However, energy surface is not quadratic. Minimum energy cannot be determined with one Newton-Raphson step. Apply the procedure iteratively. Pros: Converges quickly near a minimum (where quadratic) Requires few energy function calculations Cons: Unstable far from a minimum For large systems the inversion of Hessian becomes intractable. Large storage requirements

Second derivative method: Quasi-Newton (BFGS) Does not calculate the inverse Hessian matrix. Start with guess Hessian and update the Hessian after each step to form a more accurate Hessian. Pros: Avoids calculating the Hessian Requires few energy calculations Cons: Requires storage proportional to N 2 Inefficient in regions where the second derivative changes rapidly

First derivative method: Steepest descent Uses 1 st derivative to locate the general direction of the minimum. Uses line searching along a given direction to find structure of lower energy. The next direction of movement is orthogonal to the previous one. Example start

First derivative method: Steepest descent Pros: Does not require initial structure to be near the minimum. Good for minimizing initial structures – Relieves highest-energy features in a structure. Cons: Slow to converge (very slow at low gradient values (near minima)) Information about previous steps is lost.  Near the minimum the minimization overshoots the minimum point. Requires large number of energy evaluations.

First derivative method: Conjugate gradient Increases the efficiency by controlling the choice of new direction (Fletcher-Reeves model) Pros: Quicker to converge for large molecules Requires less iterations Cons: Unstable far from a local minimum. Requires more function evaluations in a line search (more complete) Example

When to use various minimization methods Steepest descent: used for initial minimization ( steps) Conjugate gradient, Newton-Raphson: used to complete the minimization Convergence criteria

Normal mode analysis: Vibration frequencies 1D (diatom) near stationary point >0 for minima <0 for TS frequency (in cm  1 ) real for minima imaginary for TS N-atom (3N-6 dim) for all q for other q’s for only one q (reaction coordinate) minimum: All real frequencies TS: One imaginary frequencies Characterize stationary points from the number of imaginary frequencies

for other q’s for only one q (reaction coordinate) TS: One imaginary frequencies for all q minimum: All real frequencies

Frequency and Characterizing Stationary points

Normal mode analysis: Vibration frequencies Obtain/interpret vibrational spectra (IR, Raman, HREELS, etc) Obtain zero-point energy Obtain thermodynamic quantities (enthalpy, entropy, free energy, etc.)

Vibrational Frequency Scale Factor

Geometry: An example