A unified approach to dynamic NMR based on a physical interpretation of the transition probability

Abstract

A general theory of the effect of dynamics (relaxation and (or) exchange) on NMR spectra is presented. This theory is based on a reexamination of the transition probability. The classic expression for this is as the square of the transition moment, but we feel it is useful to separate the square into two separate terms. In the generalization presented here, we show that one of these terms corresponds to the share of the initial magnetization that each spin coherence receives at the start of the experiment. The second term is how much that coherence contributes to the total detected signal. The final intensity is the product of these two factors. For a static spectrum, these two terms are complex conjugates, so the product is real and we recover the standard transition probability. When there is dynamics, the product becomes complex, so the time evolution includes oscillatory and dispersive terms. This means that a dynamic spectrum is still a sum of individual transitions, but the lineshapes are distorted in phase, intensity, position, and linewidth by the dynamic process. In this paper we develop the general theory, and illustrate it with a calculation of the classic problem of mutual exchange in an AB spin system.Key words: NMR spectroscopy, transition probability, chemical exchange, kinetics.RCsumC : On prCsente une thCorie gCnCrale de l'effet des processus dynamiques (relaxation et (ou) Cchange) sur les spectres RMN. Cette thCorie est basCe sur un rkexamen de la probabilitk de transition. L'expression classique pour ceci implique le cam6 du moment de transition; toutefois, on croit qu'il est utile de sCparer le carrC en deux termes. Dans la gCnCralisation prCsentCe ici, on montre que I'un de ces termes correspond h la partie de la magnktisation initiale que chaque coherence de spin reqoit au dCbut de l'expkrience. Le second terme est un reflet de la partie de cette cohtrence qui contribue au signal qui est dCtectC. L'intensitC finale est le produit de ces deux facteurs. Pour un spectre statique, ces deux termes sont des conjuguCs complexes et le produit est donc rCel; on rCcuptre donc la probabilitk de transition standard. Quand le spectre est dynarnique, le produit devient complexe et I'tvolution avec le temps comprend des termes oscillatoires et de dispersion. Ceci signifie qu'un spectre dynamique est encore une somme de transitions individuelles, mais, i cause du processus dynamique, les formes des bandes sont dCformCes par la phase, l'intensitt, la position et la largeur de la bande. Dans ce travail, on a dCveloppt la thtorie gCntrale et on l'a illustree h l'aide d'un calcul du problkme classique de 1'Cchange mutuel dans un systtme de spins AB.

Electronic Control of the Rotational Barrier in η2‐Alkyne‐1‐thio Complexes

“…In contrast, the rotational barrier in 9 is so low that the coalescence temperature is just reached at 220 K, while the exchange is fast at room temperature. The activation parameters of 7-9 were estimated by line-shape simulation performed with the MEXICO program package [14] (see supporting information for details). The corresponding values of 2 have already been published.…”

Operator formalisms: An overview

“…The original formulations of the Redfield (36,37,(65)(66)(67). Similarly, the standard descriptions of chemical exchange (68,69) are done in Liouville space. Including these terms (R for relaxation, X for exchange) changes Eq.…”

“…[19] is no longer Hermitian. This means that the eigenvalues and eigenvectors will have real and imaginary parts, which leads to such concepts as dynamic frequency shifts (72) and the concept of complex-valued transition probabilities in chemical exchange spectra (69). However, such formal solutions may not lead us to a satisfying picture of what is going on.…”

“…Similarly, because chemical exchange is mathematically similar to relaxation, it is much easier to treat in Liouville space. The standard approaches (68,69) are formulated in this way. Finally, angular momentum results, such as the Wigner-Eckart theorem (23), can relate many of the Liouville-space matrix elements.…”

Multinuclear (1H,31P and195Pt) NMR study and dynamical analysis of binuclear µ-hydrido µ-carbonyl Pt(i) cations with chelating diphosphines

“…K is the kinetic matrix with elements where k mn is the rate constant for the reaction from site m to site n. At equilibrium, the equation of motion simplifies to so that the transverse magnetization vector is obtained by diagonalising the complex symmetric matrix L. The complex eigenvalues of L give the frequency and width of the transitions, while the eigenvectors determine their intensity and phase. 24 The exchange models detailed above have been formalized within this theoretical framework. Some constraints have been applied: (i) the J-couplings have been assumed to be temperature independent; (ii) the homogenous line width (T Ϫ1 2,p ) has been assumed equal for all transitions in all sites; (iii) when present, all off-diagonal elements of matrix have been assumed equal.…”