I 100% echo what @cduvallet said here. You probably shouldn’t use percentile-normalized data to calculate alpha- (or even beta-) diversity metrics, without putting a lot of thought into your interpretation of the data. Random pseudo-counts to replace zeros is one issue (so metrics like Jaccard don’t make sense anymore because there are no longer zeros/absences). Another issue is that any relative abundance information within a sample is lost (i.e. the counts for each taxon are normalized to the control distribution for that taxon across samples). Thus, all percentile-transformed ‘abundances’ are numbers between 0 and 100. There’s no longer any way to distinguish what taxon is more or less abundant within a given sample. So, if this within-sample relative abundance info is necessary for the diversity metric that you are calculating, then you probably shouldn’t use percentile normalized output for that calculation.
The original application of percentile normalization is differential abundance testing on a taxon-by-taxon basis (i.e. is ‘taxon X’ enriched/depleted in cases vs. controls). It works well in this application, but extreme caution is advised for other use cases.