# Calculating Reorganisation energy using ORCA

I wondered if anyone knows of a way to calculate the reorganization energy of a molecule utilizing orca 5.0.3.

• Do you want to study the reorganization energy of electron transfer, or excitation, or intersystem crossing, or some other process? There is no such thing as the reorganization of a molecule if you don't specify what process of the molecule you are studying. Commented Aug 15, 2023 at 15:29
• And to echo @wzkchem5 question .. electron transfer (neutral <-> anion) is different from hole transfer (neutral <-> cation) Commented Aug 18, 2023 at 20:24
• 8 hours ago a similar question was asked on Reddit: reddit.com/r/comp_chem/comments/15uulyj/… Commented Aug 19, 2023 at 3:35

You essentially have two options.

The first, simpler and more precise, is the so-called 4-point strategy. As the name says, you will need a total of 4 calculations. Two geometry optimisations (one on each PES involved) and two single point calculations (each at the equilibrium geometries found before, but on the other PES). To make an example, let's say you want to compute the reorganisation energy for electron transfer. So, you optimise your neutral molecule, and your anion, obtaining their equilibrium geometries, and their energies, respectively $$E_n^n$$ and $$E_-^-$$. In this notation, the subscript denotes the PES, and the superscript the geometry, for instance.

Then, using the equilibrium geometry of the neutral molecule, you run a single point calculation setting the molecule as if it were an anion, without changing the geometry, and you obtain $$E_-^n$$. Also, using the equilibrium geometry of the anion, you run a single point as if the molecule were neutral, obtaining $$E_n^-$$. Each pair of energies on the same PES defines, through their difference, the reorganisation energy for the semi-reaction.

$$\lambda_n = E_n^- - E_n^n$$

$$\lambda_- = E_-^n - E_-^-$$

The total reorganisation energy of the process is obtained by summing the two reorganisation energies of the semi-reactions.

$$\lambda_{ET} = \lambda_n + \lambda_-$$

Obviously, both contributions should be positive, since you are subtracting from an arbitrary point on the PES the energy of the minimum. If they are not positive, you are making some mistake.

The second option, sometimes called Displaced Harmonic Oscillator (DHO), is more complicated, but can be useful, for instance when optimising a geometry on a certain state is difficult. In this case, you can exploit to the harmonic approximation, and compute the normal modes on the PES on which you were able to optimise the geometry. Then you resort to Duschinsky's equation to approximate the PES of the other electronic state. In case of our example concerning electron transfer, you get

$$Q_- = JQ_n + K$$

where $$Q_-$$ is the set of normal modes on the PES of the anion, $$Q_n$$ is the set of normal modes on the PES of the neutral molecule, $$K$$ is the vector of displacements on the PES of the anion from the equilibrium positions, and $$J$$ is a rotation called Duschinsky matrix. There are various approximations you can adopt on this equation. For instance, you can decide that $$J = I$$. In this way, it becomes easy to compute the displacement vector $$K$$ from the gradient computed on the PES of the anion at the geometry of the neutral molecule: if you think about it, you know that along each vibration you are approximating the potential with a parabola. So, knowing the slope of the tangent to that parabola tells you the displacement. In essence, you project the cartesian gradients that you get as output from a QM code onto the normal modes, to get the displacement along each mode. From the displacements, you should be able to obtain Huang-Rhys factors, or reorganisation energies along each mode. The total reorganisation energy on the PES that you are approximating this way is given by summing over modes. Of course, since you are making use of the harmonic approximation, displacements from equilibrium positions should be small. Then this approach might become difficult to adopt if (mass-weighted-)cartesian coordinates are not very appropriate for the problem, and you should use internal coordinates.

For some formulas concerning this second approach, you can have a look at this paper by Neese, to get an idea of the proper units to carry out the operations. For the approximations you can adopt on Duschinsky's equation and their appropriateness, my favourite paper is this one by Santoro. If I remember correctly, these approximations used in the second approach should be implemented in ORCA, since they are also used in the calculation of vibronic electronic spectra, resonance Raman, etc. You have to check what data they give you as output.

Finally, I used electron transfer as an example, but the same approaches can be used for hole transfer (in that case you deal with the neutral molecule and the cation), exciton transfer (in which case you deal with ground and excited state), or other processes involving any two electronic states.

• +1. Nice first answer. Welcome to our new community and we hope to see much more of you in the future!!! Thank you for your contributions! Commented Aug 19, 2023 at 1:08