How to create a differential spectrum from two simulated spectra, calculated by two different basis sets?

Question

In my bachelor thesis, I compare simulated IR-spectra computed by different basis sets. The simulated spectra from sets like cc-pVTZ and cc-pVQZ are too similar to compare them by plotting them next to each other. That's why my supervisor asks me to create a differential spectrum from these spectra. I try to subtract one from the other, but the x-axes don't align, because I scaled them with different scaling factors. When I subtract them without scaling I get a similar result. In both cases, the miss alignment of the peaks results in a jump from positive to negative values and vice versa in the differential spectra.

I wonder if there is a way to create a differential spectrum from two simulated spectra computed with different basis sets.

Edit:

To give you a better understanding of my problem I upload my spectra. All spectra are not scaled with scaling factors.

This spectrum (red) was computed with B3LYP/6-31G**.

This spectrum (blue) was computed with B3LYP/6-311++G**.

That was my try to subtract the blue spectrum from the red spectrum.

Here I divided the blue spectrum by the red spectrum like Cavenfish said.

I want to show how much higher or lower the intensity is in the red spectrum compared to the blue spectrum. In my understanding I should have intense peaks when the difference is high and peaks with low intensity when the difference between the peak intensities is small.

Answer 1

If the peaks shift inconsistently, you will never fix this with scaling factors etc. Consider if you can make a table of peak assignments with relative shifts + maximum peak heights instead. However, you can still present either of your spectra that you have calculated (subtracted or divided) if it helps you visually present an argument.

Your calculated spectra by DFT is likely not a calculated spectra, it is a calculation of excitations wavelengths and oscillator strengths. Your visualization tool such as GaussView is likely doing gaussian broadening on things, so you can get this table of data directly from your output rather than having to look on the graph. This method can also be used for UV-VIS (from TD-DFT) not just IR/Raman.

Use whatever representation best allows for you to compare things. Graphs are nice, but sometimes a table of deltas can be more descriptive to see trends. You can also make a plot of experimental expectation vs DFT results like the plot below.

Answer 2

Notice that after you divided the blue spectrum's values by the red spectrum's values, you got something that looks almost exactly like the red spectrum, but with much smaller numerical values. What you may like to do now is to use a peak alignment tool such as the alignsignals function in MATLAB's Signal Processing Toolbox or any of the many tools for aligning peaks (there's even some programs that allow you to do this in a GUI by dragging and dropping one spectrum over the other until the peaks are aligned: if you need software-recommendations on how to do this, you can ask a new question here with that tag!).

Once you have the peaks aligned, you the issue of your difference in spectra having sharp negative to positive transitions will be expected to improve significantly.

Furthermore you can "normalize" the data by dividing both spectra by the largest value in its own spectrum, which will make the tallest peak in each spectrum have the same height, but then the other peaks may be overcompensated, so whether or not you want to do this last step is up to you, and will probably depend on how satisfied you get with the result after applying my suggestion of using a peak alignment tool.