Spins with more than one J coupling For spins with several coupling partners, all couplings evolve simultaneously, but can be treated sequentially with product operators just as J coupling and chemical shift evolution) l1xcos(πJ l1xcos(πJ2t)cos(πJt) +2 Ilyl3z cos(πJ2t)sin(πJt) 2llyl2z sin(TJ12t) 2Ilyl2z sin(TJ12t cos(πJ13t) sin(πJ12t)sin(πJ1t) The double antiphase term 41y2: 13: develops straightforward from the Ily factor in 211y12- according to the normal coupling evolution rules Ily--211x13sin(/13t) When we consider the time evolution of the single antiphase terms required for coherence transfer, such as III2- Sin(T//) cos(J13t) and 21/ 3- cos(TJ/2t)sin(/13t), we find that their trigonometric factors(the transfer amplidute )al ways assume the general form 2lnl2=Sin(J/n2)cos(πJ1cos(πJ4t)cos(πJ1st with J12 being called the active coupling(that is actually responsible for the cross-peak) and all other the passive couplings When all J couplings are of the same size, then the maximum of these functions is not at t=72J, but at considerably shorter times, between ca. 16Jand /4(depending on the number of cosine factors elaxation
49 Spins with more than one J coupling For spins with several coupling partners, all couplings evolve simultaneously, but can be treated sequentially with product operators (just as J coupling and chemical shift evolution). J12 J13 I1x ¾¾® I1x cos(pJ12t) ¾¾® I1x cos(pJ12t) cos(pJ13t) + 2I1yI3z cos(pJ12t) sin(pJ13t) + 2I1yI2z sin(pJ12t) + 2I1yI2z sin(pJ12t) cos(pJ13t) - 4I1xI2z I3z sin(pJ12t) sin(pJ13t) The double antiphase term 4I1yI2z I3z develops straightforward from the I1y factor in 2I1yI2z , according to the normal coupling evolution rules I1y ¾® - 2I1xI3z sin(pJ13t). When we consider the time evolution of the single antiphase terms required for coherence transfer, such as 2I1yI2z sin(pJ12t) cos(pJ13t) and 2I1yI3z cos(pJ12t) sin(pJ13t) , we find that their trigonometric factors (the transfer amplidute) always assume the general form 2I1yI2z sin(pJ12t) cos(pJ13t) cos(pJ14t) cos(pJ15t) … with J12 being called the active coupling (that is actually responsible for the cross-peak) and all other the passive couplings. When all J couplings are of the same size, then the maximum of these functions is not at t = 1 /2J , but at considerably shorter times, between ca. 1 /6J and 1 /4J (depending on the number of cosine factors and relaxation)
1.0 sin(πJt) sin(πJt)cos(πJt) sin(πJt)cs2(πJt) 0.5 00 0. 0.020 0.040 0 0.100 -0.5 J,=12 Hz; T2=1s 1.0 However, in real spin systems the size of J varies considerably, for 'JHH from ca. 1 Hz up to ca 12 Hz(or even 16-18 Hz for J and trans in olefins ). The largest passive coupling determines when the transfer function becomes zero again for the first time(e.g,12J=35 ms for J=14 Hz), and the maximum of single antiphase coherence the occurs at or shortly before ca. /4J for this coupling constant. With only one passive coupling constant and a very small active coupling, one could wait till after the first zero passing to get more intensity. However, with a large number of passive couplings of unknown size(as in most realistic cases), the only predictable maximum will occur at 20-30 Hz for most spin systems
50 However, in real spin systems the size of J varies considerably, for 2, 3JHH from ca. 1 Hz up to ca. 12 Hz (or even 16-18 Hz for 2 J and 3 Jtrans in olefins). The largest passive coupling determines when the transfer function becomes zero again for the first time (e.g., 1 /2J = 35 ms for J = 14 Hz), and the maximum of single antiphase coherence the occurs at or shortly before ca. 1 /4J for this coupling constant. With only one passive coupling constant and a very small active coupling, one could wait till after the first zero passing to get more intensity. However, with a large number of passive couplings of unknown size (as in most realistic cases), the only predictable maximum will occur at 20-30 Hz for most spin systems