Hello
This follows on from my question on ECD spectra. I now have all the spectra but now all the computed ECD spectra has to be 'weighted using the Boltzmann distribution based on their calculated relative energies and then combined to produce the final calculated spectra.'
I would really appreciate it if someone cal explain clearly how this is to be done. I have searched for a detailed example on how to proceed but was unsuccessful.
Thanks in advance.
Hello Thishana
Here solution.
Good luck!
Ryszard
Say you have identified 3 conformations/states with relative energies E1, E2 and E3. Their Boltzmann weights will be w1 = exp(-E1/kBT), w2 = ... and the sum of weights q = w1 + w2 + w3.
Now if you have 3 different spectra A1(v), A2(v) and A3(v) corresponding to these conformations/states, your Boltzmann-weighted spectrum will be [w1 * A1(v) + w2 * A2(v) + w3 * A3(v)]/q. So just a weighted average of three functions with Boltzmann factors as weights, as simple as that.
Side note: of course the accuracy of this approach depends on whether you identified/sampled all meaningful states of the system and have no degeneracy/redundancy in your set of conformers, but I suppose you're already aware of that. For that reason, many people choose to first sample the conformational space with e.g. molecular dynamics (MD), extract a set of equally spaced frames and do calculations on that set -- since it's already implicitly Boltzmann-weighted by the MD algorithm, you can extract the final spectrum by simple (arithmetic) averaging.
I would also suggest to calculate and use the corresponding free energy values, say G1, G2, and G3, in place of E1, E2, and E3, respectively. This step would add some more accuracy in the w values, and, probably, with no need of extra calculations because free energy corrections to the energy are usually computed within vibrational frequency runs.
Hello Milosz
I have followed what you said. And yes, Alessandro, I have used the Free energy values, to calculate, w for each conformer. And I have q, which is the sum.
When you say "Now if you have 3 different spectra A1(v), A2(v) and A3(v) corresponding to these conformations/states..." what exactly does 'A1(v)' refer to?
As I said, a proper example detailing each step would be nice...
After I have a value (once you explain how to calculate it)...what/how do I use it?
Thanks
Dear Thisana,
There is the way I calculate first the populations for each conformation.
Let's say one have three conformations with G1
Dear Isabela
Unfortunately, I am still confused. I followed you up to the summing...and then I'm lost...
I have attached my Excel file. Maybe you can help.
Thanks
A1(v), A2(v) etc. are just your individual spectra of conformations 1, 2, etc. I assumed you have them stored as explicit functions of the frequency, wavenumber or any other equivalent quantity. Then it should be trivial to multiply (scale) the function by a single number (the weight) and add several weighted functions together.
The problem is, in your Excel file you only have a single CD spectrum -- how did you obtain a single set of rotary strengths from 4 computations, did you just merge them together? To be able to Boltzmann-weight them, you should have separate spectra (expressed as functions) for individual conformers (I assume they are conformers). Then you can scale these functions and add them together.
I agree that deltaG can yield a better estimate of relative populations. And, as a future note, Excel is not the best way to handle big datasets -- you might want to look into Matlab, R or Mathematica to make your working space less messy ; )
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