An Intuitively Understandable Quality Measure for Theoretical Vibrational Spectra

Computational chemistry has become a central tool in spectroscopic studies in most of chemical science. The quality of a calculated vibrational spectrum is commonly expressed as the deviation of the peak position from the experimental reference. With the increasing application of vibrational spectroscopy to complex (biological) systems, this is likely not sustainable. Here we present a quality measure for theoretical vibrational spectra based on matching the spectra to a reference database with the help of correlation coefficients. This approach can easily be applied to large sets of data and complex spectra without easily identifiable peak positions. We demonstrate this on a database of infrared spectra of 670 compounds using six different theoretical (DFT and force field) methods. Most importantly, it is intuitively understandable by both theoreticians and experimentalists.


S2 Directions of Comparison and Wavenumber Ranges
The comparison direction theoretical spectra versus experimental database (below also ,,forward comparison") means the theoretical spectrum for one compound is compared to all experimental spectra by calculating it's correlation coefficient with all experimental spectra. The theoretical spectrum is assigned to the compound with the experimental spectrum giving the highest correlation coefficient -in principle all theoretical spectra could be assigned to the same compound. In the comparison experimental spectra versus theoretical database (also ,,reverse comparison"), one experimental spectrum is compared to all all calculated spectra, correspondingly. Comparisons were done using four different wavenumber ranges: 550-3846 cm −1 (,,full"), 550-2000 cm −1 (,,medium"), 550-1650 cm −1 (,,short"), and 2000-3846 cm −1 (,,inverse"). Table S1: Numbers of compounds correctly identified using various theoretical methods in combination with different statistical measures. The labels ,,full", ,,medium", ,,short", and ,,high" refer to different spectroscopical ranges used in the calculations: 550-3846 cm −1 , 550-2000 cm −1 , 550-1650 cm −1 , and 2000-3846 cm −1 , respectively. Comparisons were made over 670 compounds in the database. Numbers in parentheses are the mean and median rank of the correct compound in the respective comparison. Level

S4 Additional Measures for Correct Matches
In total we are presenting three different measures for quantifying the performance of the different methods in matching compounds. The immediate measure for this is naturally the number of correct matches m -how many spectra were assigned to the correct compound or, for class matches, not necessarily the correct compound, but as a compound belonging to the same class as the correct compound. Dividing this number by the number of compounds in the database N , or the class n, respectively, gives the fraction of matches as F = m/N , and F = m/n, respectively. Especially in case of class matches, however, the fraction is a poor measure for the performance of a method, as it does not balance for the probability of randomly matching a compound correctly.
In order to account for this, we create a third measure, which gives the probability a random assignment of the spectra to compounds would have to create at least the present number of matches.
The probability of generating a certain number of compound matches k over the database can be expressed as where p 0 = 1/N is the elementary probability of matching any one compound correctly.
In the case of class matches, however, the situation is somewhat more intricate. We start again by defining the elementary probability p e = n/N , this time for assigning any spectrum randomly to a compound of the class in question. The probability that a certain number c of spectra is assigned to compounds of the given class is then, noting that in principle all spectra can be assigned to the same compound: However, this gives only the number of spectra that are assigned to a compound of a given class, not the number of spectra that are correctly assigned to any (i.e. not necessarily the correct one) compound of the class. The probability that of the c spectra assigned to compounds of a class S15 a certain number k ≤ c is correctly assigned to the class can be described as checking for the spectra that actually represent compounds of the class, the probability that k of these are assigned to compounds of the same class: Combining this with the expression for p c , and summing over all values for c that can give k correct matches gives: In either case, the probability of generating a certain number of correct matches is then summed over all values equal to or larger than the actual number of matches and presented as negative decadic logarithm a measure for predictive power (pP): For practical reasons, we have capped the possible values of pP at 300.0.