LETTER pubs.acs.org/JPCL
Pathway Selective Pulses Clark D. Ridge and Jamie D. Walls* Department of Chemistry, University of Miami, Coral Gables, Florida 33124, United States
bS Supporting Information ABSTRACT: Determining the availability of evolution pathways in quantum systems, which is important to a variety of spectroscopies, is often accomplished with a pathway selection scheme (PSS). In the absence of pulsed field gradients, the availability of an evolution pathway is only determined after combining the results from multiple experiments in a PSS. In this work, we provide a general method for determining the availability of evolution pathways within a single experiment by converting a PSS into a pathway selective pulse (PSP) sequence that excites the quantum system only if the selected pathways are available. Liquidstate NMR demonstrations of a PSP that nutates a spin only in the presence of heteronuclear coupling is presented. SECTION: Kinetics, Spectroscopy
C
ontrolling the evolution and/or evolution pathways that a quantum system follows is important to a variety of multidimensional spectroscopies. One common method of implementing this control is through the use of a pathway selection scheme (PSS). In a PSS, multiple experiments are performed and combined such that the observed signal arises from only a small subset of evolution pathways. The overall effect of a PSS can be interpreted as a linear transformation of the initial density matrix PSS
into a temporally averaged “density matrix”, F, i.e., Finit f F. This transformation is nonunitary, which can be understood in terms of an effective decoherence arising from the abrupt “decay” or filtering of unwanted pathways. In nuclear magnetic resonance (NMR), PSSs are often implemented by cycling the phases of the radio frequency (RF) pulses, commonly referred to as phase cycling,1 in order that only selected coherence pathways contribute to the overall signal. In optical spectroscopies, analogues to phase cycling, where pulses using different polarization and wave vectors are combined,2,3 have been used to select the signals from only a subset of nonlinear processes. PSSs are often used to determine whether certain evolution 6 0) or unavailable (F = 0) to a quantum pathways are available (F ¼ system. Such information has been used to provide important information about the coupling network, size, and structural constraints in spin systems.4 In both NMR5 and (to a lesser extent) optical spectroscopies,6 pulsed field gradients can be used to reduce the acquisition time for coherence pathway selection by imparting a spatial dependence to the phases of the various coherence pathways so that only the signal from selected coherence pathways adds constructively over the sample volume. However, in PSSs that require more than pulse phase cycling and/or situations where pulsed field gradients are not readily available, multiple experiments must be performed in order to isolate the signal from the evolution pathways of interest. In this case, there can be a r 2011 American Chemical Society
considerable cost in acquisition time for the PSS, especially in slowly relaxing or time-sensitive systems. In this Letter, we propose a general method to speed up the determination of the availability for certain evolution pathways by converting a PSS into a coherent, pathway selective pulse (PSP) that excites a system only if F 6¼ 0. When the excitation can be detected directly, the determination of availability of certain pathways can be realized in a single experiment. Liquid-state NMR demonstrations of PSPs in heteronuclear spin systems are presented. First, evolution pathways and pathway selection must be defined. Consider a closed quantum system of dimension n, described by the ^(t), where ^1 is the n-dimensional identity density matrix (1/n)^1 + F ^(t) is the nonidentity portion or deviation density matrix, and F ^(t) can be expanded matrix (i.e., Trace[^ F(t)] = 0). Assume that F ^ 2,...,O ^ M}, at ^ 1,O in an M-dimensional subset of operators, L = {O ^ ^(t) = ∑M all times, i.e., F k=1ck(t)Ok for all t. (For simplicity, assume ^ j.) In this case, the evolution under the Hamiltonian ^(t0) = O F ^ α(tN,t0)^ ^ †α(tRN, ^ α(t) simply transforms F ^(t0) to F ^(tN) = U F(t0)U H M α ^ k, where U ^ exp((i/p) ttN ^ α(tN,t0) = T t0) = ∑k=1ckj(tN,t0)O 0 0^ 0 ^ ^ dt H α(t )) is the propagator under H α(t), T is the Dyson timeα † † ^ kU ^ jU ^ α(tN,t0)O ^ α(tN, ordering operator, and ckj(tN,t0) = {Tr[O ^ †kO ^ k]} is the transition amplitude for evolving from t0)]}/{Tr[O ^ j to O ^ k under U ^ α(tN,t0). Consider the case where the prop^(t0) = O F agator can be broken up into the product of N time-ordered prop^ ΠN1 ^ ^ α(tN,t0) = T agators at times t1, t2, ..., and tN, i.e., U k=0 U α(tk+1,tk). ^j f F ^(tN) can be ^(t0) = O In the L basis, the evolution of F described in terms ofτ MN different N-step evolution pathways, τ1 τ2 N ^ m ∈ L for k = 1 to k = N and ^ ^ ^ m ... f ^ m with O Oj f Om 1 f O O 2 N k with τn = tn tn1. An example of evolution pathways for an Received: July 21, 2011 Accepted: September 12, 2011 Published: September 12, 2011 2478
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The Journal of Physical Chemistry Letters
^ 0/p = of the spin angular momentum operator. In this case, H η(pω/kBT)^I Z can be generated by a DC magnetic field applied along the +^z-direction of strength η(pω/kBT). ^ †k (tN,t0) for Assuming that the time-reversed propagators, U k = 1 to k = Nexp, can be experimentally generated, the propagator ^k U ^ k(tN,t0), is exactly equivalent ^ †k(tN,t0) P for the pulse block, U to evolving under a Hamiltonian that is proportional to the deviation ^ †k(tN,t0) = ^ k(tN,t0)P^kU density matrix of the kth experiment, i.e., U † ^ ^ ^k(tN)]. A PSP F(t0)U k(tN,t0)] = exp[iητλkF exp[iητλkU k(tN,t0)^ can be generated by concatenating the various Nexp different pulse blocks (Figure 1C). The overall propagator for the PSP in Figure 1C, ^ 123...N (where the subscript denotes the order of the pulse blocks U exp in the PSP), is given by ^ ^ 123:::Nexp ¼ T U
Nexp Y k¼1
^ ¼T
Nexp Y
^ k ðtN , t0 ÞP^k U ^ †k ðtN , t0 Þ U ^k ðtN Þ exp½iητλk F
ð1Þ
k¼1
)
)
)
^ 123...N in eq 1 and to In order to simplify the description of U exp connect the evolution under a PSP to F, average Hamiltonian 7 ^ 123...N theory can be used to rewrite the propagator as U exp (j) exp(iτH avg), where H avg = ∑∞ j=1H avg is the average Hamiltonian that the system evolves under for a time τ, and H (j) avg is the jth-order F(t0) , π/3 and Nexpτ ηF < 2π contribution to H avg. For τλkη^ (where A = (Tr[A†A])1/2 is the Frobenius matrix norm of A), H avg can be approximated8 by )
^ 1 ,O ^ 2,O ^ 3}, evolving M = 3-dimensional set of operators, L = {O under a two-step propagator is shown in Figure 1A. As shown in Figure 1A, there are 32 = 9 possible two-step pathways under ^ 2, which is transformed into ^ 1(t2,t0) starting from F ^(t0) = O U ^ k. ^1(t2) = ∑3k=1c1k2(t2,t0)O F From the above definitions, it is important to note that evolution pathways are different from coherence transfer pathways typiτ1 τ2 callyτ used τ in NMR. For example, ^I ( f ^I (^SZ f ^I ( and ^I ( f1 ^I ( f2 ^I ( shown in Figure 2A represent two distinct evolution pathways but correspond to identical coherence transfer pathways, where the system remains as single-quantum coherence during the two-step evolution. While the following paper focuses on designing PSPs for evolution pathways, the results can be easily modified to also include PSPs for coherence pathway selection. Selection of only a subset of the MN possible evolution pathways can be accomplished using a PSS by combining the results exp ^k(tN), where λk, ^Fk(tN) = from Nexp e MN with F = ∑N k=1 λkF † ^ ^ ^ U k(tN,t0)^ F(t0)U k(tN,t0), and U k(tN,t0) are the kth experiment’s weighting factor, deviation density matrix, and propagator, respectively. In Figure 1B, an example of a PSS that selects the pathway τ1 τ2 ^3 f ^ 1 is illustrated. In this case, a total of Nexp e 32 ^2 f O O O ^ k(t2,t0) for k = 1 to k = Nexp, are different two-step evolutions, U performed, and the resulting density matrices are combined exp ^ 1, where only the pathway to generate F = ∑N Fk(t2) = c1(t2,t0)O k=1 λk^ τ1 τ2 ^3 f O ^ 1 (blue) contributes to F, i.e., c1(t2,t0) = ^2 f O O k k exp λ c (t ,t )c ∑N k=1 k 13 2 1 32(t1,t0). If such a pathway was not allowed, c1 (t2,t0) = 0, and no signal would be obtained. We now outline a general method of converting a PSS into a ^ 0, can be PSP. Assume that a time-independent Hamiltonian, H ^ 0/p = η^ ^(t0), i.e., H F(t0), where generated that is proportional to F η is some proportionality constant, and define the propa^k = exp[(itk/p)H ^ 0] = ^ 0 for a time tk = λkτ as P gator under H F(t0)]. For example, the deviation density matrix for exp[iτλkη^ a spin at thermal equilibrium at a temperature T and in the presence of a strong magnetic field aligned along the +^z-axis is ^ ≈ (pω/kBT)^I Z, where ω is the larmour approximately given by F frequency, kB is Boltzmann’s constant, and ^I Z is the ^z-component
Figure 2. (A) Pathways available for evolution inτ an IS system under τ1 2 two-step evolution. The pathway, ^I ( f ^I (^SZ f ^I ( (green solid), is only available if J 6 ¼ 0. (B) Experimental results from the τ1 τ2 ^I (^SZ f ^I ( PSS in a chloroform (ν = 1.5 kHz) and acetone ^I ( f (ν = 0 Hz) solution with the propagators defined in eq 4. The spectra from these experiments after multiplication by the corresponding λk are shown along with the overall 13C-edited 1H-edited spectrum [bottom], which exhibited better than a 99% suppression of the 12C-bound protons (center peaks) and absorptive/in-phase doublets for the 13CHCl3 and OC(CH3)(13CH3) isotopomers. For each of the four experiments, the number of scans (NS) was 32 with a recycle delay (D1) of 5 s. (C) PSP sequence constructed from the experiments in eqs 4 and 5, correspond^ 4312. ing to the propagator U
) )
)
)
Figure 1. A PSP in an M = 3-dimensional space of operators, L = ^ 1,O ^ 2,O ^ 3}, under two-step evolution. (A) All 32 possible evolution {O ^ 2. ^ (t ,t ), starting from F ^(t0) = O pathways under the two-step propagator, U τ 1 1 2 0 τ2 ^2 f ^3 f O ^ 1 (blue) can be (B) Selection of the pathway O O accomplished by weighting the signals from Nexp e 9 different experiexp Fk(t2) ments, corresponding to an averaged “density matrix”, F = ∑N k=1 λk^ ^ = c1(t2,t0)O1 [c1(t2,t0) is nonzero only if the pathway in panel B is allowed]. (C) The corresponding PSP consisting of Nexp different pulse ^ †k (t2,t0) P ^k U ^ k(t2,t0) for k = 1 to k = Nexp, where P ^k = blocks, U exp[iθη^ F(t0)]. If Nexp θη^F(t0) , 2π, the PSPs propagator is given ^ 123...N ≈ exp(iτηF ) (eq 2). by U exp
LETTER
H̅ avg ≈H̅ ð1Þ avg ¼
1 Nexp τλk η^ Fk ðtN Þ ¼ ηF τ k¼1
∑
ð2Þ
^ 123...N ≈ exp(iτηF). Under the above conditions, U exp ^k U ^ k(tN,t0) ^ Although the order of the various U †k (tN,t0) P (1) pulse blocks does not affect H avg in eq 2, the order does affect 2479
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The Journal of Physical Chemistry Letters higher-order terms in H avg. It should be noted that the pulse motif of “sandwiching” a small period of evolution between a propagator and its time-reversed propagator in eq 1 has been previously used to selectively generate multiple-quantum (MQ)Hamiltonians9,10 and to study quantum chaos in Loschmidt echoes.11 While both PSPs and these earlier selective MQsequences9,10 can be used to generate MQ-Hamiltonians, PSPs can be used to control “how” the MQ-Hamiltonian is generated. For example, while both methods can be used to generate an average double-quantum (2Q)-Hamiltonian in molecules with more than two-coupled spins (I = 1/2), a PSP can be designed to generate a 2Q-Hamiltonian only in molecules with three or more coupled spins by including the step (3Q f ( 2Q in generating H avg; such a step is unavailable in a molecule with only two spins, hence ^ avg = 0 in eq 2. H There are a few necessary conditions that must be fulfilled to convert a PSS into a PSP using eq 1. First, only PSSs using real λk are allowed since F must be hermitian in eq 2. The hermiticity of F also requires that both a pathway and its corresponding “conjugate” pathway contribute to F, which is not necessary for a typical PSS. For example, while the pathway I1,Z f ^I 1,+ f ^I 1,+ ^I 2,+ f ^I 2, can be selected in a three-pulse NMR experiment with heterodyne detection (where ^I k,( = ^I k,X ( i^I k,Y are the raising/ lowering operators for the kth spin), both this pathway and its “conjugate” pathway, I1,Z f ^I 1, f ^I 1, ^I 2, f ^I 2,+, must contribute to F in the corresponding PSP. While a PSP for any PSS can be constructed as long as the above conditions are satisfied, the rest of this article will focus on PSPs derived from PSSs where F6¼ 0 only if a particular evolution pathway is allowed. If the pathway is not allowed, ^ 123...N ≈ F = 0, and hence H avg ≈ 0 (eq 2). In this case, U exp exp(iτηF) =^1 in eq 1 is incapable of exciting the system, ^ 123...N F ^ †123...N ≈ F ^(0). One class of such PSSs are ^(0)U U exp exp experiments that generate 13C-edited 1H spectra, where only those 1H (spin 1/2, natural abundance ≈ 100%) that are coupled to 13C nuclei (spin 1/2, natural abundance ≈ 1%) are observed. A variety of PSSs have been developed to generate 13C-edited 1H spectra, mostly by selecting pathways involving magnetization transfer between the 1H and 13C spin systems, such as ^S( f ^I ( in a reverse INEPT12 or ^I ( f ^S( f ^I ( in an HSQC experiment,13 where ^I(^S) = (1/2)σ^ are the 1H(13C) spin operators and σ^ = {σ^Z,σ^Y,σ^X} are the Pauli spin matrices Besides magnetization transfer between the I and S spins, 13 C-edited 1H spectra can be generated by selecting pathways containing correlations between the I and S spins that develop due to the heteronuclear scalar coupling. To illustrate this scenario, we consider, for simplicity, an IS spin system that evolves under ^ IS/p = ωI^I Z + 2πJ^I Z^SZ, where ωI is the I spin’s the Hamiltonian, H resonance offset in the doubly rotating frame, and J [in hertz (Hz)] is the scalar coupling constant between the I and S spin (ωS = 0 was chosen since only I spin coherence is considered). As illustrated in Figure 2A, initial coherence/transverse magnetization on the I spin can evolve into heteronuclear antiphase cor^ free(t) = relation due to coupling with the S spin under U ^ IS]: exp[(it/p)H ^ free ðτ1 Þ^I ( U ^ †free ðτ1 Þ U ð3Þ ¼ ^I ( expð-iωI τ1 Þ½cosðπJτ1 Þ - i2^SZ sinðπJτ1 Þ ^ ^ The antiphase correlations, I (SZ, are generated only when the I and S spins are coupled (J 6¼ 0). Since only the transverse magnetization is directly observed in an NMR experiment, the
LETTER τ1
τ2
pathway ^I ( f ^I (^SZ f ^I ( [green solid, Figure 2A] can be used to discriminate coupled versus uncoupled spins, since this pathway is unavailable for the latter. One of the simplest methods to isolate the signal from only those 1H that are coupled to 13C nuclei is by using a 13C-edited proton spin echo,12,14 where the results from two proton spin echo experiments, with and without a π-pulse applied to the S from one another. In this case, both the spin,12,14 are subtracted τ τ ^I ( f1 ^I (^SZ f2 ^I (, and the antiphase pathway, in-phase pathway, τ τ ^I ( f1 ^I (^SZ f2 ^I (^SZ (Figure 2A) contribute to the overall signal, which can lead to distortions in the spectral lineshapes and can complicate the spectral analysis. In order toτ select the signal τ2 1 arising from only the in-phase pathway ^I ( f ^I (^SZ f ^I ( in Figure 2A, the following four experiments can be used [tt = τ1 + τ2]: ^ 1 ðtt , 0Þ ¼ U ^ free ðτ2 ÞU ^ free ðτ1 Þ U ^ 2 ðtt , 0Þ ¼ U ^ free ðτ2 ÞR ^ S0 ðπÞU ^ free ðτ1 Þ U S ^ 3 ðtt , 0Þ ¼ R ^ 0 ðπÞU ^ free ðτ2 ÞU ^ free ðτ1 Þ U ^ Sπ ðπÞU ^ free ðτ2 ÞR ^ S0 ðπÞU ^ free ðτ1 Þ ^ 4 ðtt , 0Þ ¼ R U
ð4Þ
where = exp[iθ(^SX cos(ϕ) + ^SY sin(ϕ))] and = exp[iθ(^I X cos(ϕ) + ^I Y sin(ϕ))] represent rotations of phase ϕ and angle θ generated by strong, on-resonant RF pulses ^(0) = applied to the S and I spin, respectively. Starting with F ^ Iπ/2(π/2)]† = ^I X = (1/2)(^I + + ^I ), a PSS can be ^ Iπ/2(π/2)^I Z[R R constructed using the four different propagators in eq 4 to select τ τ ^I ( f1 ^I (^SZ f2 ^I ( [with λ1 = λ3 = λ2 = λ4 = 1, and resulting in F = 4 sin(πJτ1) sin(πJτ2)(^I X cos(ωItt) + ^I Y sin(ωItt))]. In Figure 2B, the spectra after application of the sequences in eq 4 [τ1 = 3.5 ms and τ2 = 1.4 ms] on a 2% chloroform [CHCl3] and 1% acetone [OC(CH3)2] solution in deuterochloroform [CDCl3] are shown. The combination of the various spectra resulted in a 99% suppression of the 12C-bound protons, which correspond to the peaks at ν ≈ 1.5 kHz (chloroform) and ν = 0 Hz (acetone) [the fact that the 12 C-bound protons were not completely zeroed is most likely due to pulse errors and/or spectrometer subtraction errors in the CYCLOPS phase cycle15]. Purely absorptive spectra for the 13C-coupled protons were observed for the isotopomers 13CHCl3 and OC(CH3)^I ^S terms in F. (13CH3)], which confirmed theτ absence of τ2 ( Z 1 In order to convert the ^I ( f ^I (^SZ f ^I ( PSS into a PSP, the time reversed propagators for the propagators in eq 4 were generated by switching τ2 and τ1 in eq 4 and using the fact that ^ Iϕ+π(π) = H ^ ISR ^ IS. In this case, ^ Iϕ(π)H R ^ Sϕ(θ) R
^ Iϕ(θ) R
^ †1 ðtt , 0Þ ¼ R ^ Iϕ ðπÞU ^ free ðτ1 ÞU ^ free ðτ2 ÞR ^ Iϕ þ π ðπÞ U ^ Iϕ ðπÞU ^ free ðτ1 ÞR ^ Sπ ðπÞU ^ free ðτ2 ÞR ^ Iϕ þ π ðπÞ ^ †2 ðtt , 0Þ ¼ R U ^ †3 ðtt , 0Þ ¼ R ^ Iϕ ðπÞU ^ free ðτ1 ÞU ^ free ðτ2 ÞR ^ Sπ ðπÞR ^ Iϕ þ π ðπÞ U ^ Iϕ ðπÞU ^ free ðτ1 ÞR ^ Sπ ðπÞU ^ free ðτ2 ÞR ^ S0 ðπÞR ^ Iϕ þ π ðπÞ ^ †4 ðtt , 0Þ ¼ R U ð5Þ For ^F(0) = ^I X, P^k = exp(iλkθ^I X) can be generated by a strong RF pulse applied to the I spin (but with θ , π/3). With the sequences in eq 4 and eq 5, the corresponding PSP is shown in ^ 4312 ≈ exp(iθF) = RωI^ t (Θ) with Θ = 4θ Figure 3C, where U I t sin(πJτ1) sin(πJτ2). The PSP in Figure 2C [with τ1 = τ2 = 2 ms, θ = π/7.6, n = 1, ϕ = 0] was applied to the same acetone/chloroform solution used in Figure 2B, and the resulting spectrum is shown in Figure 3B. In Figure 3B, the signal from the 12C bound protons was suppressed by 99.7% and 99.81% for the acetone and chloroform 2480
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Figure 3. PSP in Figure 2C applied to a 2% chloroform and 1% acetone solution in CDCl3. (A) The 1H spectrum after a π/2-acquire on a 300 MHz spectrometer (inset) and vertically zoomed by a factor of 400 so that the 13C satellites are visible. The two peaks marked by asterisks are impurities. (B) The 1H spectrum after application of the PSP in Figure 2C [τ1 = τ2 = 2 ms, θ = π/7.6, n = 1, ϕ = 0]. The 1H doublets for the 13 CHCl3 and OC(CH3)(13CH3) isotopomers were both excited inphase. In A and B, NS = 128 and D1 = 5 s.
resonances. Since simulations predict a suppression of g99.995%, this small excitation was most likely due to errors in the π-pulse and/or hardware errors (see Supporting Information, Figure S3). Compared with the π/2-acquire spectrum in Figure 3A, the 13C-coupled 1H spins were excited by a simple rotation (i.e., their signals were in-phase) by Θ ≈ π/2 for 13CHCl3 [J = 209 Hz] and Θ ≈ 5π/18 for OC(CH3)(13CH3) [J = 127 Hz]. Note that with this set of parameters, the signal from the isotopomer O13C(CH3)2 was not observed (Θ ≈ π/900 [J = 7 Hz]) but can be seen if larger τ1 and τ2 values are used (data not shown). In this case, a 13C-edited, inphase 1H spectrum can be obtained directly in a single experiment as opposed to four experiments (Figure 2B). τ 1 To τillustrate that the PSP developed for the pathway ^I ( f ^I (^SZ f2 ^I ( represents a pure rotation, the nutation curve under repeated application of the symmetrized version of the PSP in ^ 43122134, is shown Figure 2C, corresponding to the propagator U in Figure 4. The PSP [τ1 = τ2 = 1.25 ms and θ ≈ π/38.3] was applied to a 99% 13C-labeled chloroform sample [2% in CDCl3]. The chloroform spectra are all in-phase, indicating that the PSP generates a pure rotation, and the experimental nutation curve was fit by 1.013 sin(nπ/8.65) e0.0172n (solid, red curve). The slight decay of the signal is attributable to RF inhomogeneities and relaxation during the PSP. The PSP duration was 40 ms, which corresponded to an effective decay constant during the PSP of T2* ≈ 2.3 s. It should be noted that selective rotations based upon J have been previously developed, such as in the BIRD (Θ = π),16 TANGO (Θ = π/2),17 and BIG-BIRD (arbitrary Θ)18 sequences, which all consist of a few pulses with flip-angles and pulse delays determined solely by J. In contrast, the choices of τ1, τ2, and θ in the PSP are independent of J, although the overall excitation, Θ, does depend on J. Optimization of the various parameters in a PSP can be performed using optimal control methods.19 Our results indicate that the PSP used in Figure 3B provides both better suppression of the 12C-bound 1H signal and in-phase signal when compared to the BIRD,16 TANGO,17 and the 13C-edited proton spin echo experiment12,14 (see Figure S2 in the Supporting Information). In summary, we have presented a general method for using a PSS to construct a PSP where the evolution is equivalent to the evolution under a time-independent Hamiltonian proportional
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Figure 4. The nutation curve, as a function of n, obtained for a 99% 13Clabeled chloroform sample [2% in CDCl3] under application of the ^ 43122134 [τ1 = τ2 = symmetrized PSP, corresponding to the propagator U ^ 43122134 was 40 ms]. The 1.25 ms, θ ≈ π/38.3, and the cycle time for U nutation curve was fit by 1.013 sin(nπ/8.65)e0.0172n (solid, red curve). [Inset] Comparison of the π/2-acquire spectrum to the spectrum after application of the PSP with n = 4 (blue spectrum). For each n, NS = 8 and D1 = 15 s.
to F from the PSS. As such, a PSP can, in principle, be used to determine the availability of specific evolution pathways within a single experiment as opposed to Nexp e MN for a PSS. However, the number of pulse blocks in a PSP increases with the complexity of the spin system (M) and number of steps (N) in the PSS as Nexp e MN. As the length of the PSP increases, both relaxation and dynamical changes during the PSP, such as chemical exchange, must be considered. While such effects may hinder the coherent excitation of spins based upon evolution pathways, understanding how relaxation and/or dynamical changes affect the steady-state responses under PSPs could possibly provide additional information about dynamics on time scales comparable to the PSP lengths. In this case, average Liouvillian theory20 could be used to better understand the effects of relaxation during the PSPs. Besides relaxation, perfect time-reversal of the individual ex^ k, is experimentally challenging. For instance, the periments, U scalar couplings between nonidentical protons, 2π∑k
