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Broadband-decoupled proton spectroscopy
JOURNAL
OF MAGNETIC
RESONANCE
95, 132- 148 ( 199 1)
Broadband-DecoupledProton Spectroscopy PING XU,XI-LI
WV, ANDRAYFREEMAN
Department of Chemistry, Cambridge University, Cambridge CB2 IEW, England Received April 9, 199 1 Proton NMR spectra are presented in a form that incorporates chemical-shift effects but excludes spin-spin splittings, as if homonuclear broadband decoupling had been employed. By Fourier transformation of spin echoes modulated by proton-proton coupling, a two-dimensional J spectrum is obtained. A 50 ms purging pulse at the end of the evolution period suppresses antiphase product-operator terms; this changes the character of the twodimensional spin-multiplet patterns and ensures that the signals are in the pure absorption mode. In the limit of weak coupling, each multiplet pattern possessesC, symmetry and a software “symmetry filter” separates overlapping multiplets and rejects all other signal components. By the appropriate projection of the processed J spectrum, a one-dimensional spectrum is obtained with narrow singlet responses at the chemical-shift frequencies and no spin-spin splittings. The peak heights are proportional to the numbers of equivalent protons at each site. A separate spin-multiplet pattern from each site is available in the other frequency dimension (Fr ). The technique can be incorporated into more complicated experiments such as spin-lattice relaxation studies. o 1991Academic press, hc.
The key to a more wide-ranging application of proton NMR spectroscopy is simplzjicution-many molecules of interest generate excessively complicated proton spectra with crowded and overlapping multiplets. It has long been the spectroscopist’s dream to emulate carbon-l 3 NMR and reduce proton spectra to a single line from each chemical site, with no spin-spin splitting. This would greatly extend the range of molecules that could be studied while still allowing spin coupling information to be reintroduced where necessary for assignment purposes. At the same time this new format for proton spectra would facilitate experiments such as spin-lattice relaxation time measurements, where spin-spin splitting represents an unnecessary complication. Considerable effort has been devoted toward this goal, inspired by the two-dimensional proton J spectroscopy experiment of Aue et al. (I). However, a satisfactory outcome has been frustrated by the peculiar lineshape inherent in this form of spectroscopy. This superposition of two-dimensional dispersion and absorption lineshapes (the phase twist) has unfortunate properties-the 45” projection vanishes, and the long dispersion tails seriously impair resolution and show interference effects in the overlap regions (2). Most of the proposed remedies have their own inherent problems. If the absolute-value mode is used, the resolution is badly degraded; pseudo-echo weighting (3) and the “constant-time” methods (4, 5) give intensities that are modulated by the J couplings and are therefore useless for quantitative studies. A dataprocessing method (6) that searches for each two-dimensional response and subtracts the dispersion component breaks down in the presence of severe overlap or weak 0022-2364191 $3.00 Copyright 0 1991 by Academic Pres, Inc. All rights of rqmduction in any form reserved.
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signals comparable with the noise. A more recent idea is the “time-reversal” pulse ( 7) which can refocus the spin-spin coupling effects during the evolution period in twodimensional experiments. By centering the time-reversal pulse in the middle of the evolution period in a two-dimensional nuclear Overhauser experiment, a homonuclear broadband-decoupled NOESY spectrum was obtained in the I;, frequency dimension. The weakness of this scheme is the lower sensitivity and the appearance of some spurious cross peaks. Experiments such as constant-time COSY (5) and time-reversal NOESY ( 7), both of which employ a projection onto the FI axis, suffer from limited resolution because of the restriction in the number of tl increments. We propose here a new form of two-dimensional J spectrum. It can be projected onto the F2 axis to give pure absorption-mode proton signals with no spin-spin splittings and with intensities that accurately reflect the numbers of equivalent protons at each site. This is analogous to the standard form of broadband-decoupled carbon- 13 spectroscopy. Two innovations make this feasible. First, a purging pulse is inserted at the end of the evolution period of a conventional two-dimensional homonuclear J spectroscopy experiment. The effect is to suppress dispersion-mode components, leaving a J spectrum with pure absorption mode in both frequency dimensions. Furthermore, the spin-spin splitting pattern is radically altered-instead of the multiplet splitting running along a 45” diagonal, it forms a square pattern with C, symmetry, centered at the chemical-shift frequency. This is crucial to the utilization of the new technique since it permits the use of a “symmetry filter” that rejects all signal responses lacking this particular symmetry and thus disentangles overlapping spin-multiplet patterns. The processed two-dimensional spectrum closely resembles the form of a heteronucleur two-dimensional J spectrum where broadband decoupling has been applied in the F2 dimension. The projection onto the F2 axis gives the decoupled spectrum and the individual spin-multiplet patterns from each site may be extracted if necessary. TWO-DIMENSIONAL
J SPECTROSCOPY
The conventional two-dimensional J spectroscopy sequence (1) consists of a ?r/2 pulse to generate transverse magnetization and a ?r pulse at the middle of evolution period to refocus the chemical shifts, so that during the t, period the effective Hamiltonian represents only spin-spin coupling. Consider a weakly coupled system of two spin- 1 nuclei (IS). The evolution can be predicted by the product-operator formalism (8) giving a density operator for the I spin at the end of the evolution period: uI = -Z,cos(?rJt,)
+ 2I,S,sin(aJt,).
111
The terms -Jy and 2 I,& represent in-phase and antiphase magnetizations, respectively. After two-dimensional Fourier transformation, the spectrum has the spin-multiplet structure confined to 45” diagonals that intersect the F2 axis at the chemical-shift frequencies. Phase-twist lineshapes are obtained, with the disadvantage that the 45” projection vanishes, while the 45’ sections give multiplet patterns in which the small splittings are distorted by interference effects between overlapping dispersion tails. The usual remedy is to record the absolute-value mode, but although the 45” projection can now be obtained, it is unsatisfactory from the point of view of lineshapes and
134
XU, WU, AND FREEMAN
relative intensities. Figure 1 illustrates a typical two-dimensional tricyclodecanone derivative (I)
J spectrum of a
F
I
which has been sheared through 45” to bring the multiplet patterns parallel to the new F, axis. The projection onto the new F2 axis gives the decoupled spectrum. Note the distortion of lineshapes (due to the absolute-value display) and relative intensities. These problems have long been recognized but no general solution has been found to date. Note that strong coupling between protons D and E gives rise to some additional
II
f f I* I’ z
I
i0 Hz
FIG. 1. Two-dimensional Jspectrum of a tricyclcdecanone derivative obtained by the conventional method. The data table has been sheared through 45” to bring the spin-multiplet patterns parallel to the F’, axis. The projection onto the F2 axis gives undesirable lineshapes (absolute-value mode) and intensity distortions. Additional lines appear between the chemical shifts of protons D and E due to strong-coupling effects.
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SPECTROSCOPY
lines at the mean chemical shift, with splittings in the Fr dimension that are considerably larger than the coupling constants. PURGED
J SPECTROSCOPY
The key to the proposed new form of J spectroscopy is the introduction of a purging pulse at the end of the evolution period to suppress certain antiphase product-operator terms (2Z,S,) while retaining other terms (Z,,). This changes the nature of the spinmultiplet structure in the two-dimensional spectrum. Purging can be accomplished by applying a strong radiofrequency field B1 along the y axis of the rotating frame as in a spin-locking experiment (Fig. 2a), relying on the spatial inhomogeneity of B1to disperse the 2Z,S, components by differential precession about the y axis. Alternatively we may use an “adiabatic pulse” (9-Z4), which is rather more effective in the presence of instrumental imperfections. The frequency vt of the adiabatic pulse starts from the exact resonance condition and is swept at a rate that satisfies the adiabatic condition
[21 Magnetization that was initially along the y axis “follows” the effective field BeEasit tilts in the yz plane. The effective field BeEsubtends an angle 19with respect to the +z axis, where tan 81 = (-YBI)/[‘W~,
- 41;
tan OS= (yBl)lP4vt
- 41,
[31
where vI and vs are the chemical shifts of two coupled spins. The frequency sweep is halted at a point where ~9,and OSare close to the magic angle (54.7’), where it is known (9) that the spatial inhomogeneity is most effective in dephasing zero-quantum coherence. After a suitable interval (in practice 35 ms) to allow appreciable dispersion of x components by differential precession about Bea,the adiabatic sweep is resumed.
180’(Y)
900(x) Et1
a
180’(y)
9W(x)
b
spin-lock WY)
90’(Y)
Et1
FIG.2. Two alternative pulse sequences for the purged .I spectroscopy experiment. The innovation is the introduction of either (a) a spin-locking pulse or (b) an adiabatic pulse and a hard r/2 pulse at the end of the evolution period.
136
XU,
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AND
FREEMAN
One method is to reverse the sweep direction to return the surviving magnetization back toward the y axis. In practice we continued the frequency sweep away from resonance so as to carry the magnetization toward the z axis. It does not quite reach the +z axis but retains a small y component, and a hard 7r/2cY) pulse is then applied to excite x magnetization, leaving a small y component. The latter is cancelled in a two-step phase cycle, where the adiabatic pulse and the receiver are alternated in phase. This purging sequence is set out in Fig. 2b. The dephasing of single-quantum and higher-order multiple-quantum coherence is effective whatever the value of the tilt angle 8. In an ideal situation, zero-quantum coherence would not be excited in this experiment, but in practice any imbalance in the excitation of the I and S spins could generate a zero-quantum term “ZQ
= ,$+ - ut)* + (-$,/2a)*
- i((vS - Y,)* +(?&/2?r)*.
[41
The use of the adiabatic pulse is thus an insurance against the effects of instrumental imperfections; in many circumstances a spin-locking pulse along the y axis would serve equally well. Instead of simplifying the two-dimensional multiplet structure, the purging operation actually introduces more lines (responses that were cancelled in the conventional J spectrum). Each spin multiplet becomes a square pattern, where the splittings in F, and F2 are multiplicative. The magnetization Z,cos( a Jt, ) that survives the purging pulse is converted into Z,cos( aJtl) by the final a/2 pulse and continues to evolve under the chemical shift vI and the scalar spin-spin coupling J during the detection period. If quadrature detection is used, the observed signal is proportional to icos(?rJt,){exp[i2?r(~,+
$J)t2]+exp[i2?r(vI-
iJ)t2]}
=A+B+C+D,
where A = $ exp(+i?rJtl)exp[i2?r(vI
+ f J)t2]
B = f exp(+irJt,)exp[i2n(vI
- 1 J)t2]
C = $ exp(-iaJtl)exp[i2?r(vl
+ fJ)t2]
D = i exp(-hJtl)exp[i2?r(vI
- fJ)t2].
151
After two-dimensional Fourier transformation, each of the four terms in Eq. [ 5 ] gives rise to a single peak of the multiplet of spin I. The four responses have the same phase in both dimensions (cosine modulation) with all the peaks of the same intensity. The peaks appear at the coordinates {F,, Fz} = {4fJ,
vI + tJ>,
161
forming a pattern with C, symmetry centered at { F1, F2 } = { 0, vI } . If the entire J spectrum is projected onto the F2 axis we obtain the conventional coupled proton spectrum. More-complicated spin systems also have spectra with C, symmetry. Figure 3 shows six examples of this new form of spin-multiplet structure from the proton J spectrum of the tricyclodecanone derivative. When more than two coupled spins are involved there are subsidiary C, symmetry centers flanking the principal symmetry center, but
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137
SPECTROSCOPY
lb
a @)
8
0
Q
00
000
00
00 00
m 900
00 00
e0
000
oe
00
moo
00
FIG. 3. Six typical spin multiplets extracted from the two-dimensional purged J spectrum of a tricyclodecanone derivative. The responses are all in the pure absorption mode and in the same sense. Note that each multiplet pattern possessesC, symmetry.
they only represent part of the intensity of the two-dimensional multiplet. A straightforward symmetry filter would also accept some of these, giving a projected spectrum with the expected response at the chemical-shift frequency, together with undesirable “sideband” responses. Consider the example of a weakly coupled four-spin system (I&R) with I Jrs I > 1JIR 1, illustrated in Fig. 4. The I-spin spectrum has one principal symmetry center (a) at IFI, I;,>a = (0, VI> 171 together with several subsidiary symmetry centers, 12 of which (b through e) are indicated in the figure: {F,,
f&
=
(0,
VI f
JI,}
(F,,
J’dc
=
{*JIs,
VI>
{FI,
F&
=
{+JIs,
VI +
{F,,
Fz}e
=
{*&JIs,
VI +
JIS} t.4~).
[81
It is important to note that all principal C, symmetry centers must lie on the F2 axis and therefore the search is conducted only along this axis. This rules out 10 of these 12 subsidiary symmetry centers, leaving just two at IF,, F,),
= (0, VI f JIS)
[91
138
XU, WU, AND FREEMAN .I 2%
J IS
4 00
00
86
d
b
00
d
Q
0
e @
e 0
0 a
C
863
C
F,
00 e
e 0
0
0 d
00
@@ b @@
00 d 00
Fl 4. Schematic two-dimensional purged J spectrum of the I spin of an ISzR spin system. There is a principal C, symmetry center at a, and subsidiary C, symmetry centers at b, c, d, and e. Only a and b are accepted by the symmetry filter. FIG.
with only one-eighth of the intensity of the principal symmetry center (a). It is important to suppress all subsidiary centers if the simplicity of the final spectrum is to be retained. A slight modification of the search routine accomplishes this (see below). In addition there may occasionally be accidental degeneracies in the coupling constants, for example, in an IRS spin system if J ra = f Jrs, there are fortuitous local symmetry centers at {FI, F2) = {WIS, {FI, &I
VI>
= (0, 0 * @Is}.
1101
Only the latter, which lie on the Fi = 0 axis, are in danger of being passed by the symmetry filter. Accidental symmetry features are rejected by the modification designed to suppress subsidiary symmetry centers. THE SYMMETRY
FILTER
In the cases of weakly coupled systems, the F2 coordinate of the C, symmetry center represents the chemical shift. In this section we introduce a data manipulation based on C, symmetry which locates these chemical shifts even where there is extensive overlap of the two-dimensional multiplets. Figure 5 shows the data manipulation in a schematic way. As Eq. [ 51 shows, the purged J spectrum is symmetrical about the axis F, = 0. Normally one would impose C, symmetry by performing r/2, ?r, and 37r/2 rotations about the presumed symmetry center, but it is more efficient to use two consecutive mirror symmetry operations with respect to the diagonal (45” ) and the antidiagonal
BROADBAND-DECOUPLED
d
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SPECTROSCOPY
139
e
FIG. 5. Data-processing scheme. A search is conducted along the F2 axis with a 50 X 50 Hz test zone (a). Signals within this zone are symmetrized with respect to the 45” diagonal (b) and with respect to the 135” diagonal (c) and then projected onto a horizontal axis through the center (d), giving the individual multiplet pattern (e). A final projection onto the F2 axis gives a response (f) at the chemical shift.
( 135“) that pass through this center. When there is already mirror symmetry with respect to the axis Fr = 0, this operation also imposes C, symmetry. That is to say, if the data matrix is a square of side m units, I( a, b) is compared with I(b, a) for 45’ symmetrization I@, b) is compared with I(m - b, m - a) for 135” symmetrization. The symmetrization operation used here compares the absolute magnitude of the intensities at points symmetrically related with respect to the diagonal, replacing the higher intensity by the lower value ( 15). This effectively suppresses all signals that do not possess the requisite C, symmetry. Had there been overlap from an adjacent multiplet, this response would have been rejected by the symmetry filter. A step-bystep one-dimensional search is carried out along the F2 axis, examining a 50 Hz square area. After symmetrization, the integral of all signals within this square test zone has a maximum when the search reaches the center of a spin multiplet, and a peak-finding routine is used to locate this condition. For first-order coupling, this maximum corresponds to the chemical shift, and the integral itself is proportional to the number of equivalent protons at this particular site. In cases involving more than two coupled spins, this kind of search would normally discover some weaker subsidiary C, symmetry centers. This problem can be circumvented by accepting only centers with an integral above a predefined threshold, taking advantage of the fact that subsidiary centers always represent much lower intensities than the principal center. If the integral exceeds the threshold, the peak-finding routine locates the nearest local maximum in the integral and this is taken to be the F2 coordinate of the symmetry center. The entire two-dimensional spin multiplet is then temporarily subtracted from the experimental data set, thereby also removing the subsidiary symmetry centers and those arising from accidental degeneracies in the J
140
XU, WU, AND FREEMAN
values. Ideally the threshold should be chosen to correspond to just less than the intensity of a single proton (or two or three protons if we suspect that there are groups of equivalent spins present). After a complete scan through the J spectrum along the F2 axis, the threshold is decreased and the search repeated. For the best results, the process is repeated until all signals above the background noise level have been symmetrized and subtracted from the experimental data table. In practice, for samples with a single component, one or two searches with different threshold levels are often sufficient to locate all the lines. However, with samples of several components, or with spectra containing severely overlapped multiplets, an iterative search with a gradually decreasing threshold (6) may prove advantageous. During the search, each two-dimensional spin multiplet is held in a temporary data store. Figure 5d shows how it may be “collapsed” by projection onto a horizontal axis through the symmetry center. These multiplets are stored and used to create a new two-dimensional spectrum with each multiplet at the appropriate F2 frequency, exactly as in heteronuclear two-dimensional J spectroscopy. Figure 6 shows a stacked-trace display of the tricyclodecanone spectrum obtained in this manner. Projection onto the F2 axis (Fig. 5f) gives the one-dimensional broadband-decoupled proton spectrum where the intensity of each peak represents the signal integral for each chemically distinct site. We shall call this the integral display. Since this process locates the exact coordinates of the center of symmetry and assigns the entire intensity to this point, the linewidth in the integral display is only one data point wide, corresponding to the point that gives the maximum integral after symmetrization. This should not be taken to mean that the resolution of two close chemical shifts has been in any way improved. Unless two close responses are resolvable in the conventional spectrum, the symmetry test will find only a single C, center. To emphasize this important point, we can reimpose the experimental linewidth before displaying the final spectrum ( 16). One simple way to accomplish this is by repeated 1:2: 1 convolution of the final F2 integral display, giving an approximately Gaussian lineshape whose width is a function of the number of convolutions employed. Normally the search program is allowed to continue until the threshold reaches the general level of baseline noise. This noise indicates to the spectroscopist the limit of sensitivity of the method. Signals weaker than the noise will not of course be detected. We shall call this the conventional display. Figure 7 compares the integral and conventional displays of the broadband-decoupled proton spectrum of the tricyclodecanone derivative. The integral display inevitably has a higher signal-to-noise ratio. Note the relative uniformity of the intensities, which all represent a single proton. STRONG-COUPLING
EFFECTS
All the methods of homonuclear broadband decoupling have an Achilles heel-the problem of strongly coupled spins. Since we are now dealing with mixed eigenfunctions, a complete separation into individual proton responses is not feasible. Apparent chemical shifts are displaced, splittings are not necessarily equal to coupling constants, and additional peaks may appear in the two-dimensional J spectrum. Purged J spectroscopy has the same limitations. Fortunately strong coupling becomes less and less common with NMR spectra at higher magnetic fields.
BROADBAND-DECOUPLED
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141
c
FIG. 6. Stacked-trace display of the purged J spectrum of the tricyclodecanone deri vative. For clarity of reprod uction, only every sixth trace is shown.
142
XU, WU, AND FREEMAN
b 4 Pm
FIG. 7. Broadband-decoupled proton spectra of the tricyclodecanone derivative (a) as the “integral” display and (b) as the “conventional” display with artificially broadened lines. Three lines near the center of the spectrum are slightly weaker due to strong-coupling effects.
For a strongly coupled AB system, the additional responses can be predicted by a density-matrix or product-operator treatment ( 17). If quadrature detection is used, the signals acquired during the t2 interval can be divided into t\;YOclasses, “normal” peaks (N) and “extraordinary” peaks (E), where TV, = cos2(20)( 1 + sin 28)cos(?rJtl) X {exp[2r(v0
+ fJ-
D)t2] + exp[2a(r+, - $J+ D)t2]
I4
I$, = cos2(28)( 1 - sin 2b)cos(aJtl) X {exp[2?r(v0 + fJ+
D)t2] + exp[2a(v0 - fJ-
E, = +( 1 + sin 20)2sin 28 cos[2a(D X {exp[2a(v0+
X {exp[2x(v0
+ iJ+
I4
- fJ)tl]
;J-D])t2]+exp[2?r(v,,-
E,, = +cos220 sin 20 cos[2a(D
D)t2]
fJ+D)t2
118
- 1 J)tl]
D)t2] + exp[2a(vo - fJ-
D)t2]}/8
EC = -cos228 sin 28 cos[27r(D + 4 J)t,] X {exp[27r(v0 + $J-
D])t2]
+ exp[2r(v0
- $J+ D)t2]}/8
Ed = -( 1 - sin 28)2sin 28 cos[27r(D + f J)t,] X {exp[2a(v0
+ $J+ D)t2] + exp[2?r(v0 - iJ-
D)t2]}/8,
[ll]
where v. = $(vA + vg), D = 4 VW, tan 28 = J/Au, Au = VA - vg, and VA, vn are the chemical shifts (all measured in hertz). Table 1 sets out the frequency coordinates and intensities of the 8 normal and the 16 extraordinary peaks while Fig. 8 shows a density-matrix simulation of the purged J spectrum for a typical case with VA - VB = 20 Hz, JAB = 4 Hz, and a predominantly Gaussian lineshape. The normal lines in this case are much more intense than the extraordinary lines. The normal peaks correspond to those observed in a weakly coupled two-dimensional J spectrum, showing only J splittings in the F, dimension. After the C, symmetry operation, the projection onto the F, axis gives the expected J doublet, each line having the reduced intensity f cos’( 28)( 1 - sin 28). Intensity is lost because the symmetrization algorithm takes the lower value of two ordinates, and the outer lines
BROADBAND-DECOUPLED
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143
TABLE 1 The Two-Dimensional
Coordinates” {F,> Fd
Peak type Normal (NJ Normal (A$) Extra (E,) Extra (Eb) Extra (EC) Extra (Ed)
J Spectrum of an AB System Intensities
{*f J, k(D - 1 J,} {kh J, +(D + $ J)} {+@I {+@I { k(D
- 4 J), f(D - f J), k(D + f J), &(D
- 1 J)} + 1 J)} - f J)}
{ +(D + ; J), f(D + 4 .I)}
1 + sin 28)/4 1 - sin 28)/4 sin 20( 1 + sin 20)*/S +sin 20 cos22t9/8 -sin 20 cos22t9/8 -sin 28( 1 - sin 20)*/g cos220( cos*26’(
’ With respect to the point {0, Q}.
N,, are weaker than the inner lines N,. (In the strong coupling limit (28 + a/2) the intensity vanishes.) The corresponding projection onto the F2 axis gives two responses at (v. f D) hertz with the intensity cos2(2B)( 1 - sin 28). These responses are slightly displaced from the A and B chemical shifts, the apparent AB shift being overestimated in exactly the same way as in conventional spectroscopy, the error increasing with the
"0
FIG. 8. Simulated contour plot of the purged J spectrum of an AB spin system with JAB = 4 Hz, vA - ~a = 20 Hz, and a predominantly Gaussian lineshape. The intense responses near F, = 0 are the “normal” lines and those labeled a, b, c, and d are the “extraordinary” lines. After symmetrization, the projection onto the F2 axis (right margin) shows the expected responses close to vA and va and a weak “spurious” response at the mean chemical shift q, .
144
XU,
WU,
AND
FREEMAN
strength of the coupling. (Alternatively we may rewrite the symmetrization algorithm so that it takes the arithmetic mean intensity rather than the lower value; this gives projected responses with the increased intensities cos2(20). However this mode of symmetrization cannot handle overlapping multiplets as effectively as the lower-value method.) The extraordinary responses have been categorized as E,, Eb, EC, and Ed in Table 1. Note that the intensity of the EC lines is equal to that of the E,, lines in magnitude but negative in sign. The peculiarity of these responses is the inclusion of chemicalshift terms which increase the splittings in the F, dimension to the point where some lines may be aliased if the tl sampling rate is set too low. These responses may also be overlooked if the size of the two-dimensional test zone is too small. It is clear from the geometry of Fig. 8 that there is a center of C, symmetry at the coordinates {FI, I;21 = (0, ~1. 1121 The symmetrization operation leaves the intensities of lines E, and Ed unchanged (since they lie on the 45” or 135’ diagonals). The Ed lines have a negative intensity, so these eight lines contribute a total intensity at the center of symmetry: lad = 2 sin220.
iI31
The absolute magnitudes of the intensities of Eb and EC are the same; consequently the symmetrization procedure does not alter them. However, since they have opposite signs they make no overall contribution to the intensity at the symmetry center. The F2 projection is therefore a spurious response at the mean chemical shift of A and B with intensity 2 sin220. For the simulated case of Fig. 8, this response has an intensity of only 0.08. In the limit of very strong coupling this central response would possess all the intensity. In fact, it is the ABX spin system that is more generally representative than an AB case, and we have carried out both density-matrix simulations and analytical calculations of typical ABX spectra. A detailed description is beyond the scope of this paper, but we may draw some general conclusions that are reassuring. Because of strong coupling, the “broadband-decoupled” proton spectrum is distorted in the following manner: (a) the apparent A and B chemical shifts are slightly displaced, overestimating (vA - vg); (b) the intensities of the A and B responses are reduced; (c) a spurious doublet response appears, centered at the mean chemical shift ~(YA + vr,) with a splitting 1JAX + JBx 1; (d) the X response is at ux but it is flanked by some very weak satellites. These distortions increase as the coupling gets stronger. In the symmetrized two-dimensional purged J spectrum, the AB splitting remains equal to JAB, but the AX and BX splittings are only approximately equal to JAX and JBx . Note that similar observations may be made about conventional one-dimensional spectra of ABX spin systems. EXPERIMENTAL
Experiments were carried out on a Varian VXR-400 spectrometer operating at a proton frequency of 400 MHz. The nominal field strength for the adiabatic pulse was given by yB,/2?r = 4.2 kHz. The adiabatic frequency sweep was performed in a
BROADBAND-DECOUPLED
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145
stepwise manner, slowly enough to satisfy Eq. [ 21. The Varian 5 mm inverse-detection probe that was used had rather good B1 homogeneity. One consequence was that when spin locking was used there was a danger of exciting HOHAHA effects ( 18)) although these were absent with the adiabatic pulse. Good radiofrequency homogeneity also demands a relatively long-duration adiabatic pulse (50 ms). Although in principle a single scan should suffice (9), a two-step phase cycle cancels any axial peaks arising from the unbalanced quadrature detection. More complicated phase cycles, for example Exorcycle ( 19), can be introduced to remove unwanted artifacts. The data-processing procedure is summarized in the schematic diagram of Fig. 9. The experimental two-dimensional J spectrum has all lines in the pure absorption mode. A point-by-point search is carried out along the F2 axis. At each point, a 50 X 50 Hz square test zone is symmetrized, first with respect to the 45” diagonal and then with respect to the 135” diagonal. Any responses that lack C, symmetry are thereby eliminated. The integral of all signals remaining within this square region is then compared with a predetermined threshold. If it exceeds the threshold, the nearest maximum is located and taken to be a principal symmetry center. This two-dimensional spin multiplet is then subtracted from the experimental data table and held in a tem-
Start search
Does integral exceed threshold
I
No
?
*YeS
k
Find nearest maXimUm v Subtract 20 multiplet & store it Y Is 4 scan complete? *Yes
Lower the threshold FIG. 9. The data-processing scheme shown as a flow chart. Two-dimensional spin multiplets are extracted from the experimental data table one at a time and held in a temporary store. At the end of the search they are projected onto a horizontal axis through the appropriate symmetry center and used to build up a twodimensional J spectrum.
146
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AND
FREEMAN
porary store. This procedure circumvents the problems of subsidiary or accidental symmetry and helps disentangle interpenetrating two-dimensional spin multiplets. After a complete scan along the F2 axis, the search may be repeated with a lower setting of the threshold. In the general case, where the sample might contain several different components of differing concentrations, it would be necessary to continue with stepwise reductions in the threshold until the baseline noise level is reached. At this stage each response significantly above the noise level has been located, symmetrized, subtracted from the experimental data table, and stored in a separate location in the form of a 50 X 50 Hz test zone containing a single two-dimensional spin multiplet. Each multiplet is then “collapsed” by projection onto a line parallel to the F, axis through the center of symmetry. This shows the spin-multiplet structure for that particular proton site. Since there is a refocusing effect in this dimension, approximate values for the spin-spin relaxation times can be obtained from the linewidths. The two-dimensional J spectrum may be reconstructed by assembling these individual spin multiplets at the appropriate frequency coordinates on a background made up of experimental noise and residuals. This spectrum resembles a heteronuclear J spectrum with full broadband decoupling. Projection onto the F2 axis gives the integral display of the broadband-decoupled spectrum. Alternatively we may calculate the conventional display, which incorporates a line broadening equal to the instrumental linewidth This is conveniently achieved by repeated 1:2: 1 convolution of the frequencydomain data, giving an approximately Gaussian broadening. The conventional display has a signal-to-noise ratio lower than that of the integral display because the intensity is spread over the linewidth instead of concentrated into a single ordinate. The sample was a deuterochloroform solution of 9-hydroxytricyclodecan-2,5-dione. For the purposes of comparison, the (sheared) conventional two-dimensional J spectrum is shown in Fig. 1, together with its projection onto the F2 axis, giving a broadbanddecoupled spectrum with severe intensity distortions and broad lines. The purged J spectroscopy result is presented in Fig. 7. Note the simplicity of the “decoupled” spectrum and the relative uniformity of the intensities, each representing a single chemically distinct proton site. Figure 3 shows expansions of some of the twodimensional multiplets to illustrate the characteristic symmetry patterns. It should be noted that all peaks are in phase and that the intensity ratios in the two frequency dimensions are the same as those in the conventional spectrum. To demonstrate the versatility of the method, it has been applied to the problem of measuring proton spin-lattice relaxation times, using the standard inversion-recovery sequence followed by the purged J spectroscopy method described above. For this experiment the search program made only one pass along the F2 axis and the symmetrization used the arithmetic-mean method rather than the lower-value method. (This gives better uniformity in the intensities.) Figure 10 shows a series of inversionrecovery spectra obtained with the tricyclodecanone sample. After population inversion, each proton signal recovers at its own characteristic relaxation rate and the results are clear and straightforward to analyze. Even though this is a rigid molecule, the observed spin-lattice relaxation times cover a 2: 1 range (Table 2). This demonstration suggests that relaxation studies could be extended to much more complicated proton spectra by this technique, just as has been done in carbon- 13 spectroscopy.
BROADBAND-DECOUPLED
PROTON
147
SPECTROSCOPY
0.0
I
seconds
I
FIG. 10. Inversion-recovery spin-lattice relaxation measurements performed on the broadband-decoupled proton spectrum (integral display) of the tricyclodecanone derivative. The measured T, values are set out in Table 2. DISCUSSION
There are several unconventional aspects of the data processing in this experiment. The symmetrization scheme, which compares two intensities and replaces them by the lower value (15), is defensible only on the grounds that the spin multiplets should theoretically have mirror symmetry about the diagonals. A less drastic procedure is to replace both intensity ordinates by their arithmetic mean, but this does not permit a clean separation of interpenetrating multiplet patterns. However, the arithmeticmean method does have the advantage, in cases of strong coupling, that the intensities are less perturbed; in fact the usual intensity conservation rules are restored. Concentrating all the intensity of a given response at a single data location is another unconventional procedure. It increases the apparent signal-to-noise ratio but does not improve the sensitivity of the method (the ability to detect weak signals comparable with the noise). Similarly, although it narrows the linewidths, it does not improve the resolving power of the experiment. The technique has one other potential dangerthe limited extent of the test zone (50 X 50 Hz in our experiments) could miss the “extraordinary” lines that can occur in the case of strong coupling. Finally, it is important that the search program persevere through several F2 scans until the threshold is reduced to a level comparable with that of the noise; otherwise weak signal components could be lost. With these provisos, the new approach greatly simplifies proton spectra by removing the spin-spin splittings, giving single lines for all chemically distinct protons at the TABLE 2 Experimental Spin-Lattice Relaxation Times” A: 1.0 s D: 1.4s H: 1.8 s
OH: 0.8 s E: 1.2 s I: 1.7 s
’ For the tricyclodecanone derivative.
B: 0.9 s F: 1.2 s J: 1.7 s
c: 1.2 s G: 1.8 s K: 1.8 s
148
XU, WU, AND FREEMAN
chemical-shift positions, provided that the spin system under investigation is weakly coupled. The intensities are proportional to the numbers of equivalent protons at each site. ACKNOWLEDGMENT We thank Dr. James Keeler for kindly providing a preprint of his paper on the adiabatic pulse method (9) and for several helpful discussions. REFERENCES I. W. P. AUE, J. KARHAN, AND R. R. ERNST, J. Chem. 2. G. BODENHAUSEN, R. FREEMAN, R. NIEDERMEYER,
Phys. 64,4226 (1976). AND D. L. TURNER, J. Mum.
Reson.
26,
133
(1977). AND G. A. MORRIS, J. Magn. Reson. 43, 333 ( 1981). AND J. SMIDT, J. Magn. Reson. 35, 167 ( 1979). A. BAX AND R. FREEMAN, J. Magn. Reson. 44, 542 ( 198 1). A. J. SHAKA, J. KEELER, AND R. FREEMAN, J. Magn. Reson. 56,294 ( 1984). 0. W. WRENSEN, C. GRIESINGER, AND R. R. ERNST, J. Am. Chem. Sot. 107, 7779 (1985). 0. W. SORENSEN, G. W. EICH, M. H. LEVITT, G. BODENHAUSEN, AND R. R. ERNST, Prog. NMR Spectrosc. 16, 163 (1983). J. J. TITMAN, A. L. DAVIS, E. D. LAUE, AND J. KEELER, J. Mugn. Reson. 89, 176 ( 1990). J. BAUM, R. TYCKO, AND A. PINES, J. Chem. Phys. 79,4643 (1983). C. J. HARDY, W. A. EDELSTEIN, AND D. VATIS, .I Magn. Reson. 66,470 ( 1986). K. UGURBIL, M. GARWOOD, A. R. RATH, AND M. R. BENDALL, J. Mugn. Reson. 78,472 ( 1988). K. UGURBIL, M. GARWOOD, AND A. R. RATH, J. Mugn. Reson. SO, 448 ( 1988). G. TOWN AND D. ROSENFELD, J. Mugn. Reson. 89, 170 ( 1990). R. BAUMANN, G. WIDER, R. R. ERNST, AND K. WOTHRICH, J. Magn. Reson. 44,402 ( 198 1). P. Xu, X. L. WV, AND R. FREEMAN, J. Am. Chem. Sot., 13, 3596 (1991). L. E. KAY AND R. E. D. MCCLUNG, J. Magn. Reson. 77,258 ( 1988). A. BAX, D. G. DAVIES, AND S. K. SARKAR, J. Mugn. Reson. 63,230 (1985). G. B~DENHAUSEN, R. FREEMAN, AND D. L. TURNER, J. Mugn. Reson. 27,511 ( 1977).
3. A. BAX, R. FREEMAN,
4. A. BAX, A. F. MEHLKOPF, 5. 6.
7. 8.
9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.
OF MAGNETIC
RESONANCE
95, 132- 148 ( 199 1)
Broadband-DecoupledProton Spectroscopy PING XU,XI-LI
WV, ANDRAYFREEMAN
Department of Chemistry, Cambridge University, Cambridge CB2 IEW, England Received April 9, 199 1 Proton NMR spectra are presented in a form that incorporates chemical-shift effects but excludes spin-spin splittings, as if homonuclear broadband decoupling had been employed. By Fourier transformation of spin echoes modulated by proton-proton coupling, a two-dimensional J spectrum is obtained. A 50 ms purging pulse at the end of the evolution period suppresses antiphase product-operator terms; this changes the character of the twodimensional spin-multiplet patterns and ensures that the signals are in the pure absorption mode. In the limit of weak coupling, each multiplet pattern possessesC, symmetry and a software “symmetry filter” separates overlapping multiplets and rejects all other signal components. By the appropriate projection of the processed J spectrum, a one-dimensional spectrum is obtained with narrow singlet responses at the chemical-shift frequencies and no spin-spin splittings. The peak heights are proportional to the numbers of equivalent protons at each site. A separate spin-multiplet pattern from each site is available in the other frequency dimension (Fr ). The technique can be incorporated into more complicated experiments such as spin-lattice relaxation studies. o 1991Academic press, hc.
The key to a more wide-ranging application of proton NMR spectroscopy is simplzjicution-many molecules of interest generate excessively complicated proton spectra with crowded and overlapping multiplets. It has long been the spectroscopist’s dream to emulate carbon-l 3 NMR and reduce proton spectra to a single line from each chemical site, with no spin-spin splitting. This would greatly extend the range of molecules that could be studied while still allowing spin coupling information to be reintroduced where necessary for assignment purposes. At the same time this new format for proton spectra would facilitate experiments such as spin-lattice relaxation time measurements, where spin-spin splitting represents an unnecessary complication. Considerable effort has been devoted toward this goal, inspired by the two-dimensional proton J spectroscopy experiment of Aue et al. (I). However, a satisfactory outcome has been frustrated by the peculiar lineshape inherent in this form of spectroscopy. This superposition of two-dimensional dispersion and absorption lineshapes (the phase twist) has unfortunate properties-the 45” projection vanishes, and the long dispersion tails seriously impair resolution and show interference effects in the overlap regions (2). Most of the proposed remedies have their own inherent problems. If the absolute-value mode is used, the resolution is badly degraded; pseudo-echo weighting (3) and the “constant-time” methods (4, 5) give intensities that are modulated by the J couplings and are therefore useless for quantitative studies. A dataprocessing method (6) that searches for each two-dimensional response and subtracts the dispersion component breaks down in the presence of severe overlap or weak 0022-2364191 $3.00 Copyright 0 1991 by Academic Pres, Inc. All rights of rqmduction in any form reserved.
132
BROADBAND-DECOUPLED
PROTON
SPECTROSCOPY
133
signals comparable with the noise. A more recent idea is the “time-reversal” pulse ( 7) which can refocus the spin-spin coupling effects during the evolution period in twodimensional experiments. By centering the time-reversal pulse in the middle of the evolution period in a two-dimensional nuclear Overhauser experiment, a homonuclear broadband-decoupled NOESY spectrum was obtained in the I;, frequency dimension. The weakness of this scheme is the lower sensitivity and the appearance of some spurious cross peaks. Experiments such as constant-time COSY (5) and time-reversal NOESY ( 7), both of which employ a projection onto the FI axis, suffer from limited resolution because of the restriction in the number of tl increments. We propose here a new form of two-dimensional J spectrum. It can be projected onto the F2 axis to give pure absorption-mode proton signals with no spin-spin splittings and with intensities that accurately reflect the numbers of equivalent protons at each site. This is analogous to the standard form of broadband-decoupled carbon- 13 spectroscopy. Two innovations make this feasible. First, a purging pulse is inserted at the end of the evolution period of a conventional two-dimensional homonuclear J spectroscopy experiment. The effect is to suppress dispersion-mode components, leaving a J spectrum with pure absorption mode in both frequency dimensions. Furthermore, the spin-spin splitting pattern is radically altered-instead of the multiplet splitting running along a 45” diagonal, it forms a square pattern with C, symmetry, centered at the chemical-shift frequency. This is crucial to the utilization of the new technique since it permits the use of a “symmetry filter” that rejects all signal responses lacking this particular symmetry and thus disentangles overlapping spin-multiplet patterns. The processed two-dimensional spectrum closely resembles the form of a heteronucleur two-dimensional J spectrum where broadband decoupling has been applied in the F2 dimension. The projection onto the F2 axis gives the decoupled spectrum and the individual spin-multiplet patterns from each site may be extracted if necessary. TWO-DIMENSIONAL
J SPECTROSCOPY
The conventional two-dimensional J spectroscopy sequence (1) consists of a ?r/2 pulse to generate transverse magnetization and a ?r pulse at the middle of evolution period to refocus the chemical shifts, so that during the t, period the effective Hamiltonian represents only spin-spin coupling. Consider a weakly coupled system of two spin- 1 nuclei (IS). The evolution can be predicted by the product-operator formalism (8) giving a density operator for the I spin at the end of the evolution period: uI = -Z,cos(?rJt,)
+ 2I,S,sin(aJt,).
111
The terms -Jy and 2 I,& represent in-phase and antiphase magnetizations, respectively. After two-dimensional Fourier transformation, the spectrum has the spin-multiplet structure confined to 45” diagonals that intersect the F2 axis at the chemical-shift frequencies. Phase-twist lineshapes are obtained, with the disadvantage that the 45” projection vanishes, while the 45’ sections give multiplet patterns in which the small splittings are distorted by interference effects between overlapping dispersion tails. The usual remedy is to record the absolute-value mode, but although the 45” projection can now be obtained, it is unsatisfactory from the point of view of lineshapes and
134
XU, WU, AND FREEMAN
relative intensities. Figure 1 illustrates a typical two-dimensional tricyclodecanone derivative (I)
J spectrum of a
F
I
which has been sheared through 45” to bring the multiplet patterns parallel to the new F, axis. The projection onto the new F2 axis gives the decoupled spectrum. Note the distortion of lineshapes (due to the absolute-value display) and relative intensities. These problems have long been recognized but no general solution has been found to date. Note that strong coupling between protons D and E gives rise to some additional
II
f f I* I’ z
I
i0 Hz
FIG. 1. Two-dimensional Jspectrum of a tricyclcdecanone derivative obtained by the conventional method. The data table has been sheared through 45” to bring the spin-multiplet patterns parallel to the F’, axis. The projection onto the F2 axis gives undesirable lineshapes (absolute-value mode) and intensity distortions. Additional lines appear between the chemical shifts of protons D and E due to strong-coupling effects.
BROADBAND-DECOUPLED
PROTON
135
SPECTROSCOPY
lines at the mean chemical shift, with splittings in the Fr dimension that are considerably larger than the coupling constants. PURGED
J SPECTROSCOPY
The key to the proposed new form of J spectroscopy is the introduction of a purging pulse at the end of the evolution period to suppress certain antiphase product-operator terms (2Z,S,) while retaining other terms (Z,,). This changes the nature of the spinmultiplet structure in the two-dimensional spectrum. Purging can be accomplished by applying a strong radiofrequency field B1 along the y axis of the rotating frame as in a spin-locking experiment (Fig. 2a), relying on the spatial inhomogeneity of B1to disperse the 2Z,S, components by differential precession about the y axis. Alternatively we may use an “adiabatic pulse” (9-Z4), which is rather more effective in the presence of instrumental imperfections. The frequency vt of the adiabatic pulse starts from the exact resonance condition and is swept at a rate that satisfies the adiabatic condition
[21 Magnetization that was initially along the y axis “follows” the effective field BeEasit tilts in the yz plane. The effective field BeEsubtends an angle 19with respect to the +z axis, where tan 81 = (-YBI)/[‘W~,
- 41;
tan OS= (yBl)lP4vt
- 41,
[31
where vI and vs are the chemical shifts of two coupled spins. The frequency sweep is halted at a point where ~9,and OSare close to the magic angle (54.7’), where it is known (9) that the spatial inhomogeneity is most effective in dephasing zero-quantum coherence. After a suitable interval (in practice 35 ms) to allow appreciable dispersion of x components by differential precession about Bea,the adiabatic sweep is resumed.
180’(Y)
900(x) Et1
a
180’(y)
9W(x)
b
spin-lock WY)
90’(Y)
Et1
FIG.2. Two alternative pulse sequences for the purged .I spectroscopy experiment. The innovation is the introduction of either (a) a spin-locking pulse or (b) an adiabatic pulse and a hard r/2 pulse at the end of the evolution period.
136
XU,
WU,
AND
FREEMAN
One method is to reverse the sweep direction to return the surviving magnetization back toward the y axis. In practice we continued the frequency sweep away from resonance so as to carry the magnetization toward the z axis. It does not quite reach the +z axis but retains a small y component, and a hard 7r/2cY) pulse is then applied to excite x magnetization, leaving a small y component. The latter is cancelled in a two-step phase cycle, where the adiabatic pulse and the receiver are alternated in phase. This purging sequence is set out in Fig. 2b. The dephasing of single-quantum and higher-order multiple-quantum coherence is effective whatever the value of the tilt angle 8. In an ideal situation, zero-quantum coherence would not be excited in this experiment, but in practice any imbalance in the excitation of the I and S spins could generate a zero-quantum term “ZQ
= ,$+ - ut)* + (-$,/2a)*
- i((vS - Y,)* +(?&/2?r)*.
[41
The use of the adiabatic pulse is thus an insurance against the effects of instrumental imperfections; in many circumstances a spin-locking pulse along the y axis would serve equally well. Instead of simplifying the two-dimensional multiplet structure, the purging operation actually introduces more lines (responses that were cancelled in the conventional J spectrum). Each spin multiplet becomes a square pattern, where the splittings in F, and F2 are multiplicative. The magnetization Z,cos( a Jt, ) that survives the purging pulse is converted into Z,cos( aJtl) by the final a/2 pulse and continues to evolve under the chemical shift vI and the scalar spin-spin coupling J during the detection period. If quadrature detection is used, the observed signal is proportional to icos(?rJt,){exp[i2?r(~,+
$J)t2]+exp[i2?r(vI-
iJ)t2]}
=A+B+C+D,
where A = $ exp(+i?rJtl)exp[i2?r(vI
+ f J)t2]
B = f exp(+irJt,)exp[i2n(vI
- 1 J)t2]
C = $ exp(-iaJtl)exp[i2?r(vl
+ fJ)t2]
D = i exp(-hJtl)exp[i2?r(vI
- fJ)t2].
151
After two-dimensional Fourier transformation, each of the four terms in Eq. [ 5 ] gives rise to a single peak of the multiplet of spin I. The four responses have the same phase in both dimensions (cosine modulation) with all the peaks of the same intensity. The peaks appear at the coordinates {F,, Fz} = {4fJ,
vI + tJ>,
161
forming a pattern with C, symmetry centered at { F1, F2 } = { 0, vI } . If the entire J spectrum is projected onto the F2 axis we obtain the conventional coupled proton spectrum. More-complicated spin systems also have spectra with C, symmetry. Figure 3 shows six examples of this new form of spin-multiplet structure from the proton J spectrum of the tricyclodecanone derivative. When more than two coupled spins are involved there are subsidiary C, symmetry centers flanking the principal symmetry center, but
BROADBAND-DECOUPLED
PROTON
137
SPECTROSCOPY
lb
a @)
8
0
Q
00
000
00
00 00
m 900
00 00
e0
000
oe
00
moo
00
FIG. 3. Six typical spin multiplets extracted from the two-dimensional purged J spectrum of a tricyclodecanone derivative. The responses are all in the pure absorption mode and in the same sense. Note that each multiplet pattern possessesC, symmetry.
they only represent part of the intensity of the two-dimensional multiplet. A straightforward symmetry filter would also accept some of these, giving a projected spectrum with the expected response at the chemical-shift frequency, together with undesirable “sideband” responses. Consider the example of a weakly coupled four-spin system (I&R) with I Jrs I > 1JIR 1, illustrated in Fig. 4. The I-spin spectrum has one principal symmetry center (a) at IFI, I;,>a = (0, VI> 171 together with several subsidiary symmetry centers, 12 of which (b through e) are indicated in the figure: {F,,
f&
=
(0,
VI f
JI,}
(F,,
J’dc
=
{*JIs,
VI>
{FI,
F&
=
{+JIs,
VI +
{F,,
Fz}e
=
{*&JIs,
VI +
JIS} t.4~).
[81
It is important to note that all principal C, symmetry centers must lie on the F2 axis and therefore the search is conducted only along this axis. This rules out 10 of these 12 subsidiary symmetry centers, leaving just two at IF,, F,),
= (0, VI f JIS)
[91
138
XU, WU, AND FREEMAN .I 2%
J IS
4 00
00
86
d
b
00
d
Q
0
e @
e 0
0 a
C
863
C
F,
00 e
e 0
0
0 d
00
@@ b @@
00 d 00
Fl 4. Schematic two-dimensional purged J spectrum of the I spin of an ISzR spin system. There is a principal C, symmetry center at a, and subsidiary C, symmetry centers at b, c, d, and e. Only a and b are accepted by the symmetry filter. FIG.
with only one-eighth of the intensity of the principal symmetry center (a). It is important to suppress all subsidiary centers if the simplicity of the final spectrum is to be retained. A slight modification of the search routine accomplishes this (see below). In addition there may occasionally be accidental degeneracies in the coupling constants, for example, in an IRS spin system if J ra = f Jrs, there are fortuitous local symmetry centers at {FI, F2) = {WIS, {FI, &I
VI>
= (0, 0 * @Is}.
1101
Only the latter, which lie on the Fi = 0 axis, are in danger of being passed by the symmetry filter. Accidental symmetry features are rejected by the modification designed to suppress subsidiary symmetry centers. THE SYMMETRY
FILTER
In the cases of weakly coupled systems, the F2 coordinate of the C, symmetry center represents the chemical shift. In this section we introduce a data manipulation based on C, symmetry which locates these chemical shifts even where there is extensive overlap of the two-dimensional multiplets. Figure 5 shows the data manipulation in a schematic way. As Eq. [ 51 shows, the purged J spectrum is symmetrical about the axis F, = 0. Normally one would impose C, symmetry by performing r/2, ?r, and 37r/2 rotations about the presumed symmetry center, but it is more efficient to use two consecutive mirror symmetry operations with respect to the diagonal (45” ) and the antidiagonal
BROADBAND-DECOUPLED
d
PROTON
SPECTROSCOPY
139
e
FIG. 5. Data-processing scheme. A search is conducted along the F2 axis with a 50 X 50 Hz test zone (a). Signals within this zone are symmetrized with respect to the 45” diagonal (b) and with respect to the 135” diagonal (c) and then projected onto a horizontal axis through the center (d), giving the individual multiplet pattern (e). A final projection onto the F2 axis gives a response (f) at the chemical shift.
( 135“) that pass through this center. When there is already mirror symmetry with respect to the axis Fr = 0, this operation also imposes C, symmetry. That is to say, if the data matrix is a square of side m units, I( a, b) is compared with I(b, a) for 45’ symmetrization I@, b) is compared with I(m - b, m - a) for 135” symmetrization. The symmetrization operation used here compares the absolute magnitude of the intensities at points symmetrically related with respect to the diagonal, replacing the higher intensity by the lower value ( 15). This effectively suppresses all signals that do not possess the requisite C, symmetry. Had there been overlap from an adjacent multiplet, this response would have been rejected by the symmetry filter. A step-bystep one-dimensional search is carried out along the F2 axis, examining a 50 Hz square area. After symmetrization, the integral of all signals within this square test zone has a maximum when the search reaches the center of a spin multiplet, and a peak-finding routine is used to locate this condition. For first-order coupling, this maximum corresponds to the chemical shift, and the integral itself is proportional to the number of equivalent protons at this particular site. In cases involving more than two coupled spins, this kind of search would normally discover some weaker subsidiary C, symmetry centers. This problem can be circumvented by accepting only centers with an integral above a predefined threshold, taking advantage of the fact that subsidiary centers always represent much lower intensities than the principal center. If the integral exceeds the threshold, the peak-finding routine locates the nearest local maximum in the integral and this is taken to be the F2 coordinate of the symmetry center. The entire two-dimensional spin multiplet is then temporarily subtracted from the experimental data set, thereby also removing the subsidiary symmetry centers and those arising from accidental degeneracies in the J
140
XU, WU, AND FREEMAN
values. Ideally the threshold should be chosen to correspond to just less than the intensity of a single proton (or two or three protons if we suspect that there are groups of equivalent spins present). After a complete scan through the J spectrum along the F2 axis, the threshold is decreased and the search repeated. For the best results, the process is repeated until all signals above the background noise level have been symmetrized and subtracted from the experimental data table. In practice, for samples with a single component, one or two searches with different threshold levels are often sufficient to locate all the lines. However, with samples of several components, or with spectra containing severely overlapped multiplets, an iterative search with a gradually decreasing threshold (6) may prove advantageous. During the search, each two-dimensional spin multiplet is held in a temporary data store. Figure 5d shows how it may be “collapsed” by projection onto a horizontal axis through the symmetry center. These multiplets are stored and used to create a new two-dimensional spectrum with each multiplet at the appropriate F2 frequency, exactly as in heteronuclear two-dimensional J spectroscopy. Figure 6 shows a stacked-trace display of the tricyclodecanone spectrum obtained in this manner. Projection onto the F2 axis (Fig. 5f) gives the one-dimensional broadband-decoupled proton spectrum where the intensity of each peak represents the signal integral for each chemically distinct site. We shall call this the integral display. Since this process locates the exact coordinates of the center of symmetry and assigns the entire intensity to this point, the linewidth in the integral display is only one data point wide, corresponding to the point that gives the maximum integral after symmetrization. This should not be taken to mean that the resolution of two close chemical shifts has been in any way improved. Unless two close responses are resolvable in the conventional spectrum, the symmetry test will find only a single C, center. To emphasize this important point, we can reimpose the experimental linewidth before displaying the final spectrum ( 16). One simple way to accomplish this is by repeated 1:2: 1 convolution of the final F2 integral display, giving an approximately Gaussian lineshape whose width is a function of the number of convolutions employed. Normally the search program is allowed to continue until the threshold reaches the general level of baseline noise. This noise indicates to the spectroscopist the limit of sensitivity of the method. Signals weaker than the noise will not of course be detected. We shall call this the conventional display. Figure 7 compares the integral and conventional displays of the broadband-decoupled proton spectrum of the tricyclodecanone derivative. The integral display inevitably has a higher signal-to-noise ratio. Note the relative uniformity of the intensities, which all represent a single proton. STRONG-COUPLING
EFFECTS
All the methods of homonuclear broadband decoupling have an Achilles heel-the problem of strongly coupled spins. Since we are now dealing with mixed eigenfunctions, a complete separation into individual proton responses is not feasible. Apparent chemical shifts are displaced, splittings are not necessarily equal to coupling constants, and additional peaks may appear in the two-dimensional J spectrum. Purged J spectroscopy has the same limitations. Fortunately strong coupling becomes less and less common with NMR spectra at higher magnetic fields.
BROADBAND-DECOUPLED
*
PROTON
5OHz
SPECTROSCOPY
141
c
FIG. 6. Stacked-trace display of the purged J spectrum of the tricyclodecanone deri vative. For clarity of reprod uction, only every sixth trace is shown.
142
XU, WU, AND FREEMAN
b 4 Pm
FIG. 7. Broadband-decoupled proton spectra of the tricyclodecanone derivative (a) as the “integral” display and (b) as the “conventional” display with artificially broadened lines. Three lines near the center of the spectrum are slightly weaker due to strong-coupling effects.
For a strongly coupled AB system, the additional responses can be predicted by a density-matrix or product-operator treatment ( 17). If quadrature detection is used, the signals acquired during the t2 interval can be divided into t\;YOclasses, “normal” peaks (N) and “extraordinary” peaks (E), where TV, = cos2(20)( 1 + sin 28)cos(?rJtl) X {exp[2r(v0
+ fJ-
D)t2] + exp[2a(r+, - $J+ D)t2]
I4
I$, = cos2(28)( 1 - sin 2b)cos(aJtl) X {exp[2?r(v0 + fJ+
D)t2] + exp[2a(v0 - fJ-
E, = +( 1 + sin 20)2sin 28 cos[2a(D X {exp[2a(v0+
X {exp[2x(v0
+ iJ+
I4
- fJ)tl]
;J-D])t2]+exp[2?r(v,,-
E,, = +cos220 sin 20 cos[2a(D
D)t2]
fJ+D)t2
118
- 1 J)tl]
D)t2] + exp[2a(vo - fJ-
D)t2]}/8
EC = -cos228 sin 28 cos[27r(D + 4 J)t,] X {exp[27r(v0 + $J-
D])t2]
+ exp[2r(v0
- $J+ D)t2]}/8
Ed = -( 1 - sin 28)2sin 28 cos[27r(D + f J)t,] X {exp[2a(v0
+ $J+ D)t2] + exp[2?r(v0 - iJ-
D)t2]}/8,
[ll]
where v. = $(vA + vg), D = 4 VW, tan 28 = J/Au, Au = VA - vg, and VA, vn are the chemical shifts (all measured in hertz). Table 1 sets out the frequency coordinates and intensities of the 8 normal and the 16 extraordinary peaks while Fig. 8 shows a density-matrix simulation of the purged J spectrum for a typical case with VA - VB = 20 Hz, JAB = 4 Hz, and a predominantly Gaussian lineshape. The normal lines in this case are much more intense than the extraordinary lines. The normal peaks correspond to those observed in a weakly coupled two-dimensional J spectrum, showing only J splittings in the F, dimension. After the C, symmetry operation, the projection onto the F, axis gives the expected J doublet, each line having the reduced intensity f cos’( 28)( 1 - sin 28). Intensity is lost because the symmetrization algorithm takes the lower value of two ordinates, and the outer lines
BROADBAND-DECOUPLED
PROTON
SPECTROSCOPY
143
TABLE 1 The Two-Dimensional
Coordinates” {F,> Fd
Peak type Normal (NJ Normal (A$) Extra (E,) Extra (Eb) Extra (EC) Extra (Ed)
J Spectrum of an AB System Intensities
{*f J, k(D - 1 J,} {kh J, +(D + $ J)} {+@I {+@I { k(D
- 4 J), f(D - f J), k(D + f J), &(D
- 1 J)} + 1 J)} - f J)}
{ +(D + ; J), f(D + 4 .I)}
1 + sin 28)/4 1 - sin 28)/4 sin 20( 1 + sin 20)*/S +sin 20 cos22t9/8 -sin 20 cos22t9/8 -sin 28( 1 - sin 20)*/g cos220( cos*26’(
’ With respect to the point {0, Q}.
N,, are weaker than the inner lines N,. (In the strong coupling limit (28 + a/2) the intensity vanishes.) The corresponding projection onto the F2 axis gives two responses at (v. f D) hertz with the intensity cos2(2B)( 1 - sin 28). These responses are slightly displaced from the A and B chemical shifts, the apparent AB shift being overestimated in exactly the same way as in conventional spectroscopy, the error increasing with the
"0
FIG. 8. Simulated contour plot of the purged J spectrum of an AB spin system with JAB = 4 Hz, vA - ~a = 20 Hz, and a predominantly Gaussian lineshape. The intense responses near F, = 0 are the “normal” lines and those labeled a, b, c, and d are the “extraordinary” lines. After symmetrization, the projection onto the F2 axis (right margin) shows the expected responses close to vA and va and a weak “spurious” response at the mean chemical shift q, .
144
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FREEMAN
strength of the coupling. (Alternatively we may rewrite the symmetrization algorithm so that it takes the arithmetic mean intensity rather than the lower value; this gives projected responses with the increased intensities cos2(20). However this mode of symmetrization cannot handle overlapping multiplets as effectively as the lower-value method.) The extraordinary responses have been categorized as E,, Eb, EC, and Ed in Table 1. Note that the intensity of the EC lines is equal to that of the E,, lines in magnitude but negative in sign. The peculiarity of these responses is the inclusion of chemicalshift terms which increase the splittings in the F, dimension to the point where some lines may be aliased if the tl sampling rate is set too low. These responses may also be overlooked if the size of the two-dimensional test zone is too small. It is clear from the geometry of Fig. 8 that there is a center of C, symmetry at the coordinates {FI, I;21 = (0, ~1. 1121 The symmetrization operation leaves the intensities of lines E, and Ed unchanged (since they lie on the 45” or 135’ diagonals). The Ed lines have a negative intensity, so these eight lines contribute a total intensity at the center of symmetry: lad = 2 sin220.
iI31
The absolute magnitudes of the intensities of Eb and EC are the same; consequently the symmetrization procedure does not alter them. However, since they have opposite signs they make no overall contribution to the intensity at the symmetry center. The F2 projection is therefore a spurious response at the mean chemical shift of A and B with intensity 2 sin220. For the simulated case of Fig. 8, this response has an intensity of only 0.08. In the limit of very strong coupling this central response would possess all the intensity. In fact, it is the ABX spin system that is more generally representative than an AB case, and we have carried out both density-matrix simulations and analytical calculations of typical ABX spectra. A detailed description is beyond the scope of this paper, but we may draw some general conclusions that are reassuring. Because of strong coupling, the “broadband-decoupled” proton spectrum is distorted in the following manner: (a) the apparent A and B chemical shifts are slightly displaced, overestimating (vA - vg); (b) the intensities of the A and B responses are reduced; (c) a spurious doublet response appears, centered at the mean chemical shift ~(YA + vr,) with a splitting 1JAX + JBx 1; (d) the X response is at ux but it is flanked by some very weak satellites. These distortions increase as the coupling gets stronger. In the symmetrized two-dimensional purged J spectrum, the AB splitting remains equal to JAB, but the AX and BX splittings are only approximately equal to JAX and JBx . Note that similar observations may be made about conventional one-dimensional spectra of ABX spin systems. EXPERIMENTAL
Experiments were carried out on a Varian VXR-400 spectrometer operating at a proton frequency of 400 MHz. The nominal field strength for the adiabatic pulse was given by yB,/2?r = 4.2 kHz. The adiabatic frequency sweep was performed in a
BROADBAND-DECOUPLED
PROTON
SPECTROSCOPY
145
stepwise manner, slowly enough to satisfy Eq. [ 21. The Varian 5 mm inverse-detection probe that was used had rather good B1 homogeneity. One consequence was that when spin locking was used there was a danger of exciting HOHAHA effects ( 18)) although these were absent with the adiabatic pulse. Good radiofrequency homogeneity also demands a relatively long-duration adiabatic pulse (50 ms). Although in principle a single scan should suffice (9), a two-step phase cycle cancels any axial peaks arising from the unbalanced quadrature detection. More complicated phase cycles, for example Exorcycle ( 19), can be introduced to remove unwanted artifacts. The data-processing procedure is summarized in the schematic diagram of Fig. 9. The experimental two-dimensional J spectrum has all lines in the pure absorption mode. A point-by-point search is carried out along the F2 axis. At each point, a 50 X 50 Hz square test zone is symmetrized, first with respect to the 45” diagonal and then with respect to the 135” diagonal. Any responses that lack C, symmetry are thereby eliminated. The integral of all signals remaining within this square region is then compared with a predetermined threshold. If it exceeds the threshold, the nearest maximum is located and taken to be a principal symmetry center. This two-dimensional spin multiplet is then subtracted from the experimental data table and held in a tem-
Start search
Does integral exceed threshold
I
No
?
*YeS
k
Find nearest maXimUm v Subtract 20 multiplet & store it Y Is 4 scan complete? *Yes
Lower the threshold FIG. 9. The data-processing scheme shown as a flow chart. Two-dimensional spin multiplets are extracted from the experimental data table one at a time and held in a temporary store. At the end of the search they are projected onto a horizontal axis through the appropriate symmetry center and used to build up a twodimensional J spectrum.
146
XU,
WU,
AND
FREEMAN
porary store. This procedure circumvents the problems of subsidiary or accidental symmetry and helps disentangle interpenetrating two-dimensional spin multiplets. After a complete scan along the F2 axis, the search may be repeated with a lower setting of the threshold. In the general case, where the sample might contain several different components of differing concentrations, it would be necessary to continue with stepwise reductions in the threshold until the baseline noise level is reached. At this stage each response significantly above the noise level has been located, symmetrized, subtracted from the experimental data table, and stored in a separate location in the form of a 50 X 50 Hz test zone containing a single two-dimensional spin multiplet. Each multiplet is then “collapsed” by projection onto a line parallel to the F, axis through the center of symmetry. This shows the spin-multiplet structure for that particular proton site. Since there is a refocusing effect in this dimension, approximate values for the spin-spin relaxation times can be obtained from the linewidths. The two-dimensional J spectrum may be reconstructed by assembling these individual spin multiplets at the appropriate frequency coordinates on a background made up of experimental noise and residuals. This spectrum resembles a heteronuclear J spectrum with full broadband decoupling. Projection onto the F2 axis gives the integral display of the broadband-decoupled spectrum. Alternatively we may calculate the conventional display, which incorporates a line broadening equal to the instrumental linewidth This is conveniently achieved by repeated 1:2: 1 convolution of the frequencydomain data, giving an approximately Gaussian broadening. The conventional display has a signal-to-noise ratio lower than that of the integral display because the intensity is spread over the linewidth instead of concentrated into a single ordinate. The sample was a deuterochloroform solution of 9-hydroxytricyclodecan-2,5-dione. For the purposes of comparison, the (sheared) conventional two-dimensional J spectrum is shown in Fig. 1, together with its projection onto the F2 axis, giving a broadbanddecoupled spectrum with severe intensity distortions and broad lines. The purged J spectroscopy result is presented in Fig. 7. Note the simplicity of the “decoupled” spectrum and the relative uniformity of the intensities, each representing a single chemically distinct proton site. Figure 3 shows expansions of some of the twodimensional multiplets to illustrate the characteristic symmetry patterns. It should be noted that all peaks are in phase and that the intensity ratios in the two frequency dimensions are the same as those in the conventional spectrum. To demonstrate the versatility of the method, it has been applied to the problem of measuring proton spin-lattice relaxation times, using the standard inversion-recovery sequence followed by the purged J spectroscopy method described above. For this experiment the search program made only one pass along the F2 axis and the symmetrization used the arithmetic-mean method rather than the lower-value method. (This gives better uniformity in the intensities.) Figure 10 shows a series of inversionrecovery spectra obtained with the tricyclodecanone sample. After population inversion, each proton signal recovers at its own characteristic relaxation rate and the results are clear and straightforward to analyze. Even though this is a rigid molecule, the observed spin-lattice relaxation times cover a 2: 1 range (Table 2). This demonstration suggests that relaxation studies could be extended to much more complicated proton spectra by this technique, just as has been done in carbon- 13 spectroscopy.
BROADBAND-DECOUPLED
PROTON
147
SPECTROSCOPY
0.0
I
seconds
I
FIG. 10. Inversion-recovery spin-lattice relaxation measurements performed on the broadband-decoupled proton spectrum (integral display) of the tricyclodecanone derivative. The measured T, values are set out in Table 2. DISCUSSION
There are several unconventional aspects of the data processing in this experiment. The symmetrization scheme, which compares two intensities and replaces them by the lower value (15), is defensible only on the grounds that the spin multiplets should theoretically have mirror symmetry about the diagonals. A less drastic procedure is to replace both intensity ordinates by their arithmetic mean, but this does not permit a clean separation of interpenetrating multiplet patterns. However, the arithmeticmean method does have the advantage, in cases of strong coupling, that the intensities are less perturbed; in fact the usual intensity conservation rules are restored. Concentrating all the intensity of a given response at a single data location is another unconventional procedure. It increases the apparent signal-to-noise ratio but does not improve the sensitivity of the method (the ability to detect weak signals comparable with the noise). Similarly, although it narrows the linewidths, it does not improve the resolving power of the experiment. The technique has one other potential dangerthe limited extent of the test zone (50 X 50 Hz in our experiments) could miss the “extraordinary” lines that can occur in the case of strong coupling. Finally, it is important that the search program persevere through several F2 scans until the threshold is reduced to a level comparable with that of the noise; otherwise weak signal components could be lost. With these provisos, the new approach greatly simplifies proton spectra by removing the spin-spin splittings, giving single lines for all chemically distinct protons at the TABLE 2 Experimental Spin-Lattice Relaxation Times” A: 1.0 s D: 1.4s H: 1.8 s
OH: 0.8 s E: 1.2 s I: 1.7 s
’ For the tricyclodecanone derivative.
B: 0.9 s F: 1.2 s J: 1.7 s
c: 1.2 s G: 1.8 s K: 1.8 s
148
XU, WU, AND FREEMAN
chemical-shift positions, provided that the spin system under investigation is weakly coupled. The intensities are proportional to the numbers of equivalent protons at each site. ACKNOWLEDGMENT We thank Dr. James Keeler for kindly providing a preprint of his paper on the adiabatic pulse method (9) and for several helpful discussions. REFERENCES I. W. P. AUE, J. KARHAN, AND R. R. ERNST, J. Chem. 2. G. BODENHAUSEN, R. FREEMAN, R. NIEDERMEYER,
Phys. 64,4226 (1976). AND D. L. TURNER, J. Mum.
Reson.
26,
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(1977). AND G. A. MORRIS, J. Magn. Reson. 43, 333 ( 1981). AND J. SMIDT, J. Magn. Reson. 35, 167 ( 1979). A. BAX AND R. FREEMAN, J. Magn. Reson. 44, 542 ( 198 1). A. J. SHAKA, J. KEELER, AND R. FREEMAN, J. Magn. Reson. 56,294 ( 1984). 0. W. WRENSEN, C. GRIESINGER, AND R. R. ERNST, J. Am. Chem. Sot. 107, 7779 (1985). 0. W. SORENSEN, G. W. EICH, M. H. LEVITT, G. BODENHAUSEN, AND R. R. ERNST, Prog. NMR Spectrosc. 16, 163 (1983). J. J. TITMAN, A. L. DAVIS, E. D. LAUE, AND J. KEELER, J. Mugn. Reson. 89, 176 ( 1990). J. BAUM, R. TYCKO, AND A. PINES, J. Chem. Phys. 79,4643 (1983). C. J. HARDY, W. A. EDELSTEIN, AND D. VATIS, .I Magn. Reson. 66,470 ( 1986). K. UGURBIL, M. GARWOOD, A. R. RATH, AND M. R. BENDALL, J. Mugn. Reson. 78,472 ( 1988). K. UGURBIL, M. GARWOOD, AND A. R. RATH, J. Mugn. Reson. SO, 448 ( 1988). G. TOWN AND D. ROSENFELD, J. Mugn. Reson. 89, 170 ( 1990). R. BAUMANN, G. WIDER, R. R. ERNST, AND K. WOTHRICH, J. Magn. Reson. 44,402 ( 198 1). P. Xu, X. L. WV, AND R. FREEMAN, J. Am. Chem. Sot., 13, 3596 (1991). L. E. KAY AND R. E. D. MCCLUNG, J. Magn. Reson. 77,258 ( 1988). A. BAX, D. G. DAVIES, AND S. K. SARKAR, J. Mugn. Reson. 63,230 (1985). G. B~DENHAUSEN, R. FREEMAN, AND D. L. TURNER, J. Mugn. Reson. 27,511 ( 1977).
3. A. BAX, R. FREEMAN,
4. A. BAX, A. F. MEHLKOPF, 5. 6.
7. 8.
9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.