I do not understand why the indices of S, i.e.

d/dc

Here, * indicates complex conjugation. Differentiating should leave Sc as this part is constant. I assume the summation over all

*kl*, change to*pl*, upon differentiation:d/dc

_{p}* (**c**^{†}**Sc**) = d/dc_{p}* ("sum from*k*to*N*" "sum from*l*to*N*" c_{k*}S_{kl}c_{l}) = "Sum from*k*to*l*" S_{pl}c_{l}Here, * indicates complex conjugation. Differentiating should leave Sc as this part is constant. I assume the summation over all

*k*coefficients and wavefunctions disappears because c*, which is the differentiated variable, pairs with all wavefunctions of state*k*, and as it exits the expression upon differentiation,*p*is used instead to denote those wavefunctions whereas*k*now would be misleading. However, I do not think that the index*p*for the variable is chosen randomly.
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