Geometry Optimization. Minimization Method

• Non-derivative minimization method- Simplex

• Derivative Minimization MethodFirst 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 1st and 2nd derivatives of energy

True only near equilibrium structureOtherwise, 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

1st derivative (gradient): 3N-dim. vector

2nd derivative (hessian): 3N3N 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 1st and 2nd 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 N2

• Inefficient in regions where the second derivative changes rapidly

First derivative method: Steepest descent

• Uses 1st 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 (10-100 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 cm1)

real for minimaimaginary for TS

N-atom(3N-6 dim)

for all q

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