Optimization

Geometry Optimization

Potential Energy Surface (PES)

Molecular geometry optimization is minimization of energy of molecule to find its stable form. We first consider the potential energy surface of typical molecule as shown in Figure 1. So the minimization of energy getting the coordinate for structure let assumed in Cartesian coordinate (XYZ format) as following equation 1.



Figure 1. Potential ENergy Surface (PES)

Energy Minimization

The Born-Oppenheimer approximation is used to reduce the complexity cost of computation. The geometry of a molecule at zero absolute temperature (temp = 273.5 K) corresponds to the minimum of the total energy:

coordinate (x,y,z) = min (< \psi | \hat(H) | \psi(*) >) Eq 1.

Where wave function and its twin are conjugated each other and H is Hamiltonian operator. Therefore, the process of finding the coordinates that minimize the energy is known in the trade as molecular geometry optimization in which It is usually the first step in the calculation of the flowchart displayed in Figure 2.


We know that the steepest ascent direction of any pathway is called gradient which is 1st derivative of energy. Also the 1st derivative of gradient is Hessian that we would consider it as a judgement to identify the state of molecule.

G_i = d(E_i)/d(r_i) Eq 2.

H_ij = d(E_ij)/[d(r_i) d(r_j)] Eq 3.


If the gradient is equal to 0 (zero) and with another condition that 

if H can be evaluated -->
   then All H eigenvalues positive => Minimum point
   else All H eigenvalues negative => Maximum point
   else  H eigenvalues of both signs => Saddle point (transition state)
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Minimization method:

  1. Nelder-Mead simplex
  2. Gradient descent: steepest ascent
  3. Conjugated gradients
  4. Newton-Raphson method

Figure 2. Geometry optimization flowchart

References

  1. Ilya Kuprov, University of Southampton, 2012
  2. J. Nocedal, S.J. Wright, Numerical Optimization, Springer, 2006.
  3. J.B. Foresman, A. Frisch, Exploring Chemistry with Electronic Structure Methods, Gaussian, 1996.