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We know that Reklt (Imklt) depends only on Im pq (Repq), for which the sum of the two indices, p q, is of opposite parity to that of k l. Thereby if initially only Rekl elements with even sum indices are nonzero, among all the elements of the time derivative of the density matrix only Imklt with odd sum indices will be nonzero. As a result, Imkl with odd sum indices will be nonzero after one step of simulation and it will only induce nonzero Reklt with even sum indices. Following this iteration, Rekl with odd sum indices and Imkl with even sum indices will be always zero when the number of investigated molecules is even or very large.
See supplementary material at http://dx.doi.org/10.1063/1.3604561 E-JCPSA6-134-029125 for details in Tables S1 and S2.
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