Geometry Optimization and Normal Coordinates
Minimize EFF by varying the geometry. (Find minimum energy geometry.)
Consider the 3N Cartesian coordinates, {x1, y1, z1, x2, y2, z2 … xN, yN, zN} or
{x1, x2, x3… x3N}, trying to find geometry for which (d EFF / d xi) = 0 for all i = 1…3N
As all the components of EFF depend either on the coordinates of individual atoms or on coordinates of pairs of atoms
EFF = ∑i ∑j EFF (xi, xj) i and j = 1 … 3N
Could write the energy as a matrix:
and then the total energy is the sum of all the terms. (Note
There is a set of coordinates {q1, q2, q3… q3N} which are related to the Cartesian coordinates {x1, x2, x3… x3N} such that:
these are the Normal Coordinates.
EFF = ∑i EFF (qi, qi)
6 of the EFF (qi, qi) = 0 (translation and rotation)
Any square, real symmetric matrix such as (EFF (xi, xj)) can by made ‘diagonal’ by a matrix A:
(Aij-1) (EFF (xi, xj))(Aij) = (EFF (qi, qj))
and this matrix relates the normal coordinates to the Cartesian coordinates:
Even better is that the minimum energy geometry is given by (d EFF / d qi) = 0 for all i, which becomes: (d EFF(qi,qi) / d qi) = 0.
(d EFF(qi,qi) / d qi) = the force along the coordinate qi , indicates how much the geometry should be moved in the direction of qi
1)Read in geometry – convert to Cartesian Coordinates {xi }
2)Use the Molecular Force Field model to calculate all EFF (xi, xj)
3)Diagonalize matrix (EFF (xi, xj)) to get normal coordinates and total energy
4)Calculate forces along normal coordinates – if all forces are zero output this geometry and energy and then STOP
5)Change geometry to reduce the forces – new {xi }
6)Return to 2
Molecular Potential Energy Surfaces
A plot of EFF vs {qi} is (apart from being physically impossible if there are more than 2 normal coordinates) is called a Potential Energy Surface. It carries considerable information about the molecule and its chemistry.
Example: O3, fixed angle (to reduce number of normal coordinates to 2), plot of energy against the two bond lengths:
O3 O2 + O
‘channel’
minimum energy geometry
H2O, energy plotted against bond angle and both bond lengths – the symmetric stretch motion
(no dissociation evident)
A typical diatomic + atom reaction PES
AB +C ABC A + BC
E vs R(AB) and R(BC)
reaction coordinate
Characterization of PES
In the last example, AB +C ABC A + BC, could plot E vs. reaction coordinate
plot,
Usual diagram where ABC may be an intermediate or a transition state.
A minimum, (d EFF / d qi) = 0 for all i, corresponds to reactants, products, intermediates and transition states.
Can distinguish intermediates and transition states the ‘double differential’
(d2 EFF / d qid qj) > 0 for all ij pairs then the geometry is a stable minimum.
If one just one of the (d2 EFF / d qid qj) < 0 then it is necessary that i = j, the geometry is a ‘saddle point’ and qiis the reaction coordinate.
Chemically the interesting positions on the PES are:
global minimum, local minima, saddle points.
A real molecular problem involves E vs (3N-6) qi – cannot plot diagrams, have to rely on mathematical methods to search the PES to find the important positions. (Applied mathematics and Computer Science – optimization theory .)