Since the state probabilities are given by the square magnitudes of the superposition coefficients, the probability of the 13 C 2 pair being in the singlet state falls from unity to the value of cos 2 ∆Ωτ at time t = τ . Simultaneously the probability of the 13 C 2 pair being in the central triplet state increases from zero to sin 2 ∆Ωτ at time t = τ . Now consider a sub-ensemble of Sq 2-molecules, all protonated simultaneously at time t = 0, and all in the nuclear singlet state. We may ask what the state of this sub-ensemble is at a later time t = T , where T is long compared to the lifetime τ SqH -of a protonated squarate molecule, but sufficiently short that is unlikely that a squarate would become protonated during this time, and also short compared to the background relaxation rate constant R 0 S . The probability that one of these SqH -molecules remains protonated for an interval τ , and is then deprotonated during the following small interval δτ , is given by k d exp(-k d τ )δτ , through equation 38. The probability that an individual SqH -molecule has that particular history, and is still in the singlet state after deprotonation, is obtained by multiplying that factor by cos 2 ∆Ωτ . The probability of any member of the sub-ensemble still being in the singlet state at the end of the interval T is obtained by integrating over all possible values of the protonation duration τ . This gives