1）
The formula for Shannon-Wiener index is: H=-∑(Pi)(㏑Pi)
H=information content of the sample (Peter/individual)=diversity index of the community, Pi=the proportion of individuals belonging to the ith species in the sample, if the total number of individuals in the sample is N, and the number of individuals of the ith species is ni, then Pi= ni/N
The Shannon index is derived from information entropy. The larger the Shannon index, the greater the uncertainty. The greater the uncertainty, the more unknown factors in this community, that is, the higher the diversity.
The Shannon-Weiner index is used to describe the disorder and uncertainty of individuals of a species. The higher the uncertainty, the higher the diversity. Two factors are included in the Shannon-Wiener diversity index: (1) the number of species, that is, richness; (2) the equitability or evenness of the distribution of individuals in the species . A large number of species increases diversity; similarly, an increase in the uniformity of distribution of individuals among species increases diversity.
2）
Simpson's Diversity Index = probability that two randomly sampled individuals belong to different species = 1 - probability that two randomly sampled individuals belong to the same species
Assuming 
that the proportion of the number of individuals in the species to the total number of individuals in the community is , then the joint probability of 
randomly selecting two individuals is 
If we combine the probabilities of all species in the community, we get the Simpson index 
, which is
In the formula,
is the number of species. The lowest value of the Simpson's Diversity Index is 0 and the highest value is 0 
. In the former case, all individuals belong to one species, and in the latter case, each individual belongs to a different species.
Rare species play a smaller role in the Simpson Diversity Index, while common species play a larger role. The community species diversity estimated by this method requires more samples.
3）
The Chao1 index is based on the assumption that in a random sampling of a population, when rare species (singletons) are still being discovered, it indicates that some rare species remain undiscovered; until all species have been sampled to at least two doubletons, it means that no new species will be discovered.
The classic formula of Chao1 is as follows:
Sobs represents the number of species observed in the sample. F1 and F2 represent the number of singletons and doubletons, respectively.
There is another formula for correcting the deviation of Chao1 index, which is also mentioned on scikit-bio[1] (Note: QIIME uses scikit-bio), as follows:
It can be seen from the classic formula that when doubletons is 0 (that is, F2 is 0), the calculated result is meaningless, and the revised formula can solve this problem.
Chao1 index is an indicator used to reflect species richness. It infers a theoretical richness from the observed results, which is closer to the true richness. Generally speaking, the species richness that can be observed will definitely be less than the actual, so how big is the gap between the two? The answer given by the chao1 index is (F1^2)/(2*F2), which is reasonably calculated by singletons and doubletons. Analyzing the second half of the chao1 index (F1^2)/(2*F2), it is not difficult to find that its weight on singletons is higher than that of doubletons (that is, F1^2 changes faster than 2*F2).
Chao1 is a measure of species richness. It has nothing to do with abundance and evenness, but it is sensitive to rare species.