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:
 E x , x  E x , x

E x , x 
E x , x   



 E x , x 
FF
FF
FF
i
1
1
2
FF
1
2
  E x , x  
FF
1
3N
3N


 E x , x
j
FF
1
1
FF
3N
3N





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:
 E q , q

0
E q , q   


0

FF
FF
1
1
1

0
E q , q
FF
2

2

1

0



 E q , q
FF
3N
3N






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:
A

A
 q 


A
11
q q
1
21
2
3N
3N1
A  A




  A
12
13 N
3N 3N
 x

 x
 

 x
1
2
3N






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
Read in geometry – convert to Cartesian Coordinates { xi }
Use the Molecular Force Field model to calculate all EFF (xi, xj)
Diagonalize matrix (EFF (xi, xj)) to get normal coordinates and total energy
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
1)
2)
3)
4)
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 qi d qj) > 0 for all ij pairs then the
geometry is a stable minimum.
If one just one of the (d2 EFF / d qi d qj) < 0 then it is necessary that i = j, the geometry is a
‘saddle point’ and qi is 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 .)