Energy Calculations¶

Things you need to know/consider:¶

Also the ionisation of electrons

The potential energy surface is an highly dimensional surface that depicts the energy landscape of a molecule
It can be characterised by a few features:

Where all second derivatives are +ve
Local minima are where we fine isomers, conformers etc
The global is the most stable form of the molecule
- This can be defined as any change in the geometry causing an increase in energy (\(E_{(R)}\))

Where all second derivatives are -ve
- This can be defined as any change in the geometry causing an increase in energy (\(E_{(R)}\))

These stationary points can initially be found by looking at where the first derivative is 0
The plane of second derivatives is called the hessian
The equilibrium of a structure will be determined by the statistical interpretation of all the minima, given a certain temperature

We can also look at the PES and compare it to the PES of excited states
Excited states will often have different geometric/PES properties, an can be used for generation of reaction intermediates in photochemistry
In 3D, we can visualise the transition between excited and non excited states, to see how the geometry behaves and transitions
Notice the subtle difference in bond length between excited states in these PES curves

The optimisation is said to be complete when the four key properties have converged.
They are:
- Maximum force
- RMS force
- Maximum displacement
- RMS displacement
Force is the first derivative of the PES and the threshold is to make sure that in all dimensions the derivatives are within a certain distance of each other
Displacement is the energy distance between the current geometry (\(r\)) and the next step (\(r+1\))
Geometry certainty:
- Energy - 6 d.p. for Ha and 1-2 d.p. for KJ/mol
- Bond length - 3 d.p.
- Bond angle - 2 d.p.
- Dipole moment - 2 d.p.
- Polarisability - 2 d.p.
- Rotational constant - 6 d.p.

Have you converged?
- Check the four criteria (four “YES”)
- Look for normal termination
Was the minima and saddle point found?
- Check with a freq calculation
  - All frequencies should be positive
  - All but one frequency means the saddle point has been found
Is this the local or global minima?
- Run a PES scan to verify the geometry

Optimisation will only find the local minima of the initial geometry.
- Semi-empirical methods are an effective way of ensuring that the starting geometry is close to the global minima
- A PES scan run in logical locations can also help to identify any possible lower minima
Optimisation will only account for the specified geometry.
- Tautomer, isomer and conformer variance will not be accounted for and to truly represent the molecule, one needs to calculate all of these and take a weighted average
- This is particularly important for gas phase experimental results which will typically be run at higher temperatures and will thus have higher isomeric variance
Zero Point Energy is another important consideration, as the energy minimisation will find the lowest possible configuration of the geometry, but in reality, the energy within the vacuum of space is enough to prevent the molecule from resting in its minima.
- What this means in reality is that the difference between the PES minima and the actual minima is the zero point energy contribution, which can be calculated with freq

A PES scan freezes all the geometry in one conformation and modifies one single property at a time (dihedral, angle, bond length)
The energy is plotted at each change of that property to form a curve.
Running a PES scan on multiple properties produces a PES surface
It is possible to run a “relaxed” PES scan in which the geometry is optimised with each change of the geometric property.

The PES shows the thermochemical landscape of the molecule, as the differences in energy between each point on the PES directly relates the thermodynamics of the molecule
If the reactant goes from a low energy isomer to a high energy one; it’s endothermic. From high to low; exothermic