Pulse-Sequence Optimization with Analytical Derivatives. Application to Deuterium Decoupling in Oriented Phases

JOURNAL OF MAGNETIC RESONANCE, ARTICLE NO.

Series A 121, 167–177 (1996)

0157

Pulse-Sequence Optimization with Analytical Derivatives. Application to Deuterium Decoupling in Oriented Phases T. O. LEVANTE, T. BREMI,

AND

R. R. ERNST

Laboratorium fu¨r Physikalische Chemie, Eidgeno¨ssische Technische Hochschule, 8092 Zu¨rich, Switzerland Received April 18, 1996

A pulse-sequence optimization procedure that uses a gradientsearch method with analytical derivatives is demonstrated. It can be implemented for arbitrarily complex spin-system Hamiltonians and arbitrary pulse sequences. The computational method presented is time-efficient and more accurate than difference methods. It is applied to the optimization of a pulse sequence proposed by K. V. Schenker, D. Suter, and A. Pines [J. Magn. Reson. 73, 99 (1987)] for deuterium decoupling in oriented phases. The optimized sequence is tested experimentally on pentadeuterobenzene dissolved in nematic phase. q 1996 Academic Press, Inc.

INTRODUCTION

The success of modern NMR depends to a considerable extent on sophisticated pulse sequences. Sequences, such as MLEV-16, WALTZ-16, or GARP-1 for heteronuclear decoupling of spin-12 nuclei in solution (1–3), MLEV-17, WALTZ-17, or DIPSI-2 for offset-independent spin locking (4–6), and COMARO-2 and COMARO-4 (7, 8) or TPPM (9) for dipolar decoupling in solids, became indispensable tools in the routine NMR laboratory. Several sequences have their origin in a conceptual idea, leading to a prototype pulse sequence which is subsequently optimized by a systematic computer-controlled variation of the relevant parameter values. In this way, the performance can be improved, sometimes by orders of magnitude. For computer optimization of pulse sequences, numerous optimization algorithms are available (10). Often the procedure is started by a grid search or random screening for the most promising area in the multidimensional parameter space. Educated guesses can also be quite helpful in this stage of optimization. But intuition and luck are of considerable importance, too. In a second step, the optimum parameter set within the preselected volume is determined by a systematic search procedure. Strategies, such as the simplex or the conjugategradient method, can be used at this stage. In cases where a single-minimum surface is expected, a gradient-search method is not only simple but also efficient, and it is advisable to terminate any search procedure with a local gradient search for fine tuning of the optimum set of parameter values.

This paper is devoted to the implementation of gradientsearch methods in situations where an elaborate quantummechanical propagator is responsible for the time evolution induced by the pulse sequence to be optimized. In this situation, the computational expenditure can become quite staggering. The most computer-time-intensive step is invariably the computation of the local gradients in the multidimensional parameter space with at least n / 1 evaluations of the quality function at each step of optimization if the difference method is used to compute the derivatives. We describe in this paper an approach that uses analytical derivatives even for arbitrarily complex Hamiltonians of the spin system. This can significantly reduce the computation time, as computing all n derivatives will be only three times as expensive as computing the function value itself irrespective of the dimension n. This approach takes less computer time and is more accurate than the often-used difference method (10). The higher the dimension of the parameter space, the larger the time saving using the proposed procedure. We apply the gradient-optimization procedure for the optimization of a deuterium-decoupling pulse sequence, proposed by Schenker, Suter, and Pines (7, 11), in an anisotropic medium where the deuterium resonance is split by a strong quadrupolar interaction. The experiments will concentrate on proton resonance of partially deuterated, oriented molecules dissolved in a nematic liquid-crystalline phase. Applications to partially deuterated solid materials are also conceivable. Proton resonance of partially deuterated oriented molecules is of practical importance for the selective measurement of proton–proton dipolar interactions in situations where the dipolar coupling network for a fully protonated molecule or solid would be too complicated to be resolved. To further improve resolution, it is necessary to broadband decouple the deuterium spins during the proton observation. It is well known that efficient deuterium decoupling is possible by strong monochromatic irradiation of the doublequantum transition (12, 13). However, the efficiency is strongly dependent on accurate frequency matching which

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can not be satisfied simultaneously for all deuterium resonances in a multisite system. Decoupling of heteronuclear spin-12 systems is usually performed using a train of composite p pulses. Previous approaches to improve the deuteriumdecoupling efficiency were based on the same strategy, using composite p pulses designed for spin-1 systems (14–16). Of particular relevance for the present study is the pulse sequence proposed by Schenker et al. (7, 11), called COMARO-2. It serves as a prototype sequence for deriving initial sets of parameter values for the gradient-optimization procedure. COMPUTATION OF ANALYTICAL DERIVATIVES OF THE PULSE-SEQUENCE PROPAGATOR

For the numerical optimization of a pulse sequence, represented by the vector of relevant parameters p, a quality function q(p) is required to characterize the performance of the pulse sequence. The quality function is a function of the propagator U(p) and can be expressed by the eigenvalues lV l (p) and the eigenvectors ÉlV l (p) … of the average Hamiltonian HV (p) under the influence of the pulse sequence. The gradient method requires the derivatives of the quality function with respect to each parameter pj k . They will be obtained from the derivatives of the eigenvalues and eigenvectors of the average Hamiltonian. The latter are related to the derivatives of the matrix elements of the total propagator with respect to the parameters, for which explicit formulas are given. The propagator U(p) of the pulse sequence is related to the average Hamiltonian HV (p) by U(p) Å exp{ 0i HU (p)ttot (p)},

[1]

where ttot is the duration of one cycle of the pulse sequence. From the relation between the eigenvalues ll (p) of the propagator and the eigenvalues lV l (p) of the average Hamiltonian, ll (p) Å exp{ 0ilU l (p)ttot (p)},

Å

d d ll (p) Éll (p) … / ll (p) Éll (p) … . dpj k dpj k

[5]

We multiply both sides of Eq. [5] from the left by » ll (p)É and take advantage of the fact that the derivative (d/dpj k )Éll (p) … is orthogonal to Éll (p) … , provided that Éll (p) … is normalized (to 1) irrespective of p to find d ll (p) dU(p) Å » ll (p)É Éll (p) … , dpj k dpj k

[6]

which expresses the fact that the derivatives of the eigenvalues of U(p) are equal to the diagonal elements of the derivative of U(p) in the eigenbasis of U(p). To obtain the derivatives of the simultaneous eigenvectors Éll (p) … of the propagator U(p) and of the average Hamiltonian HV (p), we multiply Eq. [5] from the left with » lm (p)É, m x l, and find after a short calculation

» lm (p)É

d Éll (p) … Å dpj k

dU(p) Éll (p) … dpj k . ll (p) 0 lm (p)

» lm (p)É

[7]

The pulse sequence to be optimized in the following consists of n propagation intervals with or without radio frequency, and the propagator U can be written as a product of the individual pulse propagators Uj : U Å UnUn 01Un 01rrr U2U1

[8]

Uj Å exp( 0i Hj tj ).

[9]

with

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H j Å H0 / vxj Fx / vyj Fy ,

[3]

The derivative dttot (p)/dpj k is 1 if pj k is the duration of a pulse and 0 if pj k is the intensity of a pulse. The double indexing of the parameters pj k will become clear in the context of Eqs. [11] – [13]. To compute the derivatives d ll (p)/dpj k , we start from the eigenvalue equation

/

dU(p) d Éll (p) … / U(p) Éll (p) … dpj k dpj k

The time-independent Hamiltonians of the individual pulses are given in the form

i lU l (p) dttot (p) d lU l (p) d ll (p) Å 0 . dpj k ll (p)ttot (p) dpj k ttot (p) dpj k

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[4]

and equate the derivatives of the left and right sides:

[2]

one finds the derivatives

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U(p)Éll (p) … Å ll (p)Éll (p) …

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[10]

where H0 is the unperturbed system’s Hamiltonian. Fx Å (r Irx and Fy Å (r Iry are the total-angular-momentum operators, and the orthogonal components vx j and vy j express amplitude and phase of the RF pulses. The relevant parameters for the optimization of the pulse sequence are the pulse durations tj , the amplitudes vx j and vy j , and possibly the pulse radio frequencies.

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TABLE 1 One Cycle of the Optimized Decoupling Sequence TL36 Pulse flip angles in degrees: Pulse phases in degrees:

117.18 61.3

80.73 178.2

36.18 00.5

125.01 89.4

79.02 268.0

43.92 88.4

103.86 1.0

81.27 179.7

47.79 1.0

93.15 92.1

84.60 271.2

39.60 90.0

92.16 02.1

70.29 179.5

16.29 0.0

105.48 87.6

80.28 267.7

52.83 91.1

117.18 178.7

80.73 1.8

36.18 180.5

125.01 90.6

79.02 272.0

43.92 91.6

103.86 179.0

81.27 0.3

47.79 179.0

93.15 87.9

84.60 268.8

39.60 90.0

92.16 182.1

70.29 0.5

16.29 180.0

105.48 92.4

80.28 272.3

52.83 88.9

The derivatives with respect to the two types of parameters pj1 Å tj , and pj2 Å vx j and pj3 Å vy j , needed for the evaluation of Eqs. [6] and [7] are dU dUj Å Unrrr Uj/1 Uj01rrr U1 dpj k dpj k

[11]

where Éjl … and jl are eigenvectors and eigenvalues, respectively, of the operator A: AÉjl … Å jlÉjl … . The derivatives of the eigenvalues, Eq. [3], and eigenvectors, Eq. [7] of the average Hamiltonian HV (p) are the basis for the optimization. The quality function q(p) depends on the special situation and will be described in the following for a specific example.

with dUj dUj Å Å 0i Hj Uj dpj1 dtj

[12]

and dUj dUj Å dpj2 dvxj Å

Å

d exp{ 0i[ H0 / ( vx j / v )Fx / vyj Fy ]tj } dv d exp{ 0i( H j / vFx )tj } dv

Z

Z

vÅ0

[13]

H (t) Å HI / HS / HSS / HIS / HS,RF (t)

d A/xB » jlÉ Éjm … e dx

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HI Å vI Iz

[16]

HS Å vS Sz

[17] 2 z

2 3

HSS Å vQ (S 0 1)

[18]

HIS Å IDH S,

[19]

v (t) Å 0gSB1 (t)

if jl Å jm

e jl 0 e jm otherwise, jl 0 jm

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[15]

with

HS,RF (t) Å v(t){Sx cos f(t) / Sy sin f(t)}

» jlÉ BÉjm …e jl » jlÉ BÉjm …

The formalism presented so far will now be applied to the optimization of a pulse sequence for deuterium decoupling in an anisotropic medium. We consider a system with one spin I Å 12 (e.g., proton) and one spin S Å 1 (e.g., deuteron). The Hamiltonian H (t) of this system is composed of the Zeeman terms of the two spins, HS and HI , of the quadrupolar Hamiltonian of the S spin, HSS , and of the mutual interaction between the two spins, HIS . Additionally taking the RF irradiation HS,RF (t) on the S spins into account, the total Hamiltonian H (t) is given by

vÅ0

and analogously for the derivative with respect to pj3 . For the explicit calculation of the derivative in Eq. [13] the following relation, derived by Aizu (17), can be used: The derivative with respect to x for an exponential function of the sum of two noncommuting operators A and xB is given in terms of the matrix elements in the eigenbasis of A by

Å

OPTIMIZATION OF PULSE SEQUENCES FOR DEUTERIUM DECOUPLING

[20] [21]

where vI and vS are the Larmor frequencies of the respective spins, vQ is the quadrupolar coupling constant of the S spin,

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FIG. 1. (a) Graphical representation of one cycle of the optimized pulse sequence TL36. The numerical values are given in Table 1. (b) Corresponding offset dependence of the eigenvalues of the average Hamiltonian for the subspace HV / . Since only differences are relevant, the three curves have been shifted to zero average energy.

v(t) and f(t) are the amplitude and phase of the RF pulse ˜ is the second-rank Cartesian tensor of the sequence, and D two-spin interaction. For the heteronuclear spin system, the interaction HIS is given by the dipolar and the J coupling. In the high-field approximation, the interaction Hamiltonian can be truncated to

HIS Å dIS Iz Sz ,

[23]

Because Iz commutes with the Hamiltonian, we can compute separately the eigenvectors and eigenvalues in two subspaces with the Hamiltonians H / and H 0 :

H / (t) Å vS Sz / 12vI1 / 12 dIS Sz / vQ (S 2z 0 231)

Optimization Criteria Efficient decoupling of the I spins from the S spins implies that the intensity in the I-spin spectrum is mainly concentrated in a single line. An examination of the Hamiltonian H (t) of Eq. [15] shows that it commutes with Iz . It is there-

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Int( vlm ) Å É» lU m/ É I /ÉlU l0 …É2 .

[22]

where a possible scalar J coupling has been incorporated into the heteronuclear coupling constant dIS without loss of generality.

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fore possible to characterize its eigenstates with the quantum numbers mI Å {12 and to designate them as ÉlU 1/ … , ÉlU 2/ … , ÉlU 3/ … , ÉlU 10 … , ÉlU 20 … , ÉlU 30 … . The I-spin transitions occur between states with different quantum numbers mI , and nine lines can be observed in the spectrum at the frequencies vlm Å lU m/ 0 lU l0 with the intensities Int( vlm ):

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/ v(t){Sx cos f(t) / Sy sin f(t)}

[24]

H 0 (t) Å vS Sz 0 12vI1 0 12 dIS Sz / vQ (S 2z 0 231) / v(t){Sx cos f(t) / Sy sin f(t)}.

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FIG. 2. Offset dependence of the eigenvalues of the average Hamiltonian for one subspace, e.g., HV / : (top) for continuous-wave decoupling; (middle) for COMARO-4; (bottom) for TL36. Since only differences are relevant, the three curves have been shifted to zero average energy.

The chemical shift vI is irrelevant for discussing the decoupling efficiency and we may assume vI Å 0. In this case, tr( HV { ) Å (l lU l{ Å 0, leading to a zero mean frequency for all lines,

∑ vlm Å 0,

[26]

lm

as well as to a zero mean for the three strong lines only:

∑ vll Å 0.

[27]

l

There are, in principle, two ways to obtain an optimally decoupled spectrum with a single line:

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1. All nine lines coincide. All vlm are zero and therefore all eigenvalues lU l{ vanish. This cannot be met for vQ , vS , or dIS x 0 and will be disregarded. 2. The three strong transitions v11 , v22 , and v33 coincide and the other six lines have vanishing intensities. The two Hamiltonians H / (t) and H 0 (t) differ only by the term dIS Sz and it is possible to unify them in the form

H (t, V { ) Å V { Sz / vQ (S 2z 0 231) / HRF (t) [28] with V { Å vS { 12 dIS . We can now formulate the two criteria, which must be combined for determining an optimal decoupling sequence: (a) The eigenvalues lU l (V, p) of the time-averaged Ham-

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FIG. 3. Deuterium-coupled 400 MHz proton spectra of pentadeuterobenzene (C6HD5 ) in the nematic liquid-crystalline phase N4 at 290 and 300 K.

iltonian HV ( V ) resulting from H (t, V ) shall have minimum variation in an interval Vmin õ V õ Vmax , where Vmin and Vmax are chosen such that the expected parameter values V { are in the interval. (b) The eigenvectors ÉlU l (V, p)… of the average Hamiltonian shall show minimal variation in the same interval. Quality Function We define a quality function consisting of three parts, the variation of the eigenvalues, qlV (p), the variation of the eigenvectors, qÉlV … (p) of the average Hamiltonian HV (p), and a part representing the experimental constraints, qlim (p), q(p) Å qlU (p) / qÉlU … (p) / qlim (p),

[29]

where p represents the vector of the free parameters. To measure the variation of the eigenvalues, we evaluate them for a set of m different offsets Vq , Vmin õ Vq õ Vmax , and sum the variances for the three eigenvalues:

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qlU (p) 3

Å

m

1 ∑ ∑ m l Å1 q Å1

F

m

lU l ( Vq , p) 0

1 ∑ lU l ( Vr , p) m rÅ1

G

2

.

[30]

The gradients of qlV (p), expressed by the derivatives d[qlV (p)]/dpj k , can be computed as described in the section on computation of analytical derivatives, above. To measure the angular variation of the eigenvectors ÉlU l (Vr , p)… , it is sufficient to compute, for all values Vr , the scalar product with a reference eigenvector ÉlU l (Vs , p)… . Vs is conveniently chosen in the center of the interval, Vs Å (Vmax 0 Vmin)/2. The deviations of the absolute squared products from unity are measures for the angular variation and the intensity of the corresponding line. The use of the absolute squared product is necessary, because of the arbitrary phase factors of the eigenvectors. The deviations are summed for all values Vr in the interval and for the three eigenvectors l Å 1, 2, and 3. This leads then to 3

m

qÉlU … (p) Å ∑ ∑ (1 0 É» lU l ( Vr , p)ÉlU l ( Vs , p) …É2 ). l Å1 rÅ1

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[31]

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FIG. 4. 61.4 MHz deuterium resonance spectra of pentadeuterobenzene in the nematic liquid-crystalline phase N4 at 290 and 300 K.

Because of the normalization of the eigenvectors to 1, all terms of this expression remain positive. q To keep the cycle time ttot and the pulse amplitudes vj Å v 2xj / v 2yj within bounds, a constraining function qlim (p) was introduced: qlim (p) n

Å K1[ttot (p) 0 twish ] 2 / K2

∑ ( vj 0 vwish ) 2 . [32] jÅ1

COMPUTATIONS AND RESULTS

As a starting point for the optimization of the deuteriumdecoupling sequences, COMARO-type sequences (7, 11) were selected. So far, they were the best sequences known.

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The COMARO-2 phase cycle was thereby combined with all the composite-pulse elements listed in Table 2 of Ref. (7). Although COMARO-4 performs slightly better than COMARO-2 (7), we selected COMARO-2 with 36 pulses for the optimization because its sequence is shorter by a factor of two. However, the performance of the final sequence is compared with COMARO-4. Twelve different starting sequences for each composite-pulse element with the COMARO-2 phase cycle were generated by randomly varying the pulse lengths by up to 20%. To check the possibility of even better sequences, we started independent optimization runs from 300 randomly chosen sequences, also with 36 pulses in which all parameters were selected randomly. The optimization of a sequence with 36 pulses (108 parameters) required between 20 and 40 gradient-iteration

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FIG. 5. Deuterium-decoupled 400 MHz proton spectra of pentadeuterobenzene (C6HD5 ) in the nematic liquid-crystalline phase N4 at 290 K as a function of the deuterium-decoupling-frequency offset. Continuous-wave irradiation (top), COMARO-4 decoupling (middle), and TL36 decoupling (bottom).

steps. For the evaluation of the propagators and the function values needed for the gradient-search procedure, the GAMMA toolkit (18) was used. Each step took about 20 minutes on a SPARC2 workstation. We used m Å 20 and a range of {1000 Hz for the offsets in Eqs. [30] and [31]. We optimized the sequences independently for vQ /2p equal to 6, 7, 8, 9, and 10 kHz, and for a B1-field strength of 4 and 5 kHz. The cycle time twish in Eq. [32] was chosen to be 1.5 ms and K1 was set to 10 4 . The field strength vwish / 2p was selected to be either 4 or 5 kHz and K2 Å 2.5 1 10 07 was used. A subset of 13 sequences emerged from the optimization

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runs for which the value of q(p) dropped below 5 1 10 05 . The corresponding starting sequences consisted of two random sequences, and sequences originating from lines 1, 3, and 4 of Table 2 in Ref. (7). These sequences were subsequently tested experimentally. The results are discussed under Experimental. Within this ensemble, seven sequences had values q(p) below 5 1 10 06 . They all originated from random modifications of sequences using line 4 from Table 2 in Ref. (7) and are thus based on the composite-pulse structure introduced by Levitt et al. (16). The minimum outstanding value of the quality function q(p) Å 2.95 1 10 07 was reached by a pulse sequence which will be termed

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FIG. 6. Deuterium-decoupled 400 MHz proton spectra of pentadeuterobenzene (C6HD5 ) in the nematic liquid-crystalline phase N4 at 300 K as a function of the deuterium-decoupling-frequency offset. Continuous-wave irradiation (top), COMARO-4 decoupling (middle), and TL36 decoupling (bottom).

‘‘TL36.’’ This value of the quality function is so low that a further numerical optimization is not feasible, due to the limited numerical accuracy. TL36 is given in Table 1 and depicted in Fig. 1a. In Fig. 1b, the offset dependence of the energy levels of the average Hamiltonian of one subspace, HV / or HV 0 , of the new sequence is displayed. Figure 2 compares the three cases CW, COMARO-4, and TL36 decoupling within an offset range of {1000 Hz. The figures clearly show the superior performance of the optimized sequence in comparison to COMARO-4 decoupling for the conditions considered here. EXPERIMENTAL

To test experimentally the performance of the optimized pulse sequence TL36, we investigated the deuterium-decou-

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pled proton NMR spectrum of pentadeuterobenzene (C6HD5 ). Pentadeuterobenzene was dissolved in the nematic liquid-crystalline phase N4 [eutectic mixture of the two isomers of p-methoxy-p *-butylazobenzene (Licristal, Merck Art. No. 10105.0005)] at a concentration of 6.2 wt%. A standard 5 mm broadband high-resolution probehead and a 5 mm sample tube with a filling height of 45 mm were used. All spectra shown were measured overnight to minimize the external disturbances (tramways, etc.) in the absence of a field-frequency lock on a Bruker DMX 400 MHz wide-bore system. The temperature was set to 290 and 300 K for two different data sets to test the dependence of the decoupling efficiency on the quadrupolar and dipolar coupling strengths. The RF field strength on the deuterium channel was fixed at 4 kHz. The 1H acquisition after a p /2 pulse lasted for

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FIG. 7. Deuterium-decoupled 400 MHz proton spectra of pentadeuterobenzene (C6HD5 ) in the nematic liquid-crystalline phase N4 at 290 K as a function of the deuterium-decoupling-frequency offset. The range of offsets up to 1 kHz illustrates the large span (between 300 and 1000 Hz) where the peak intensities are constant within a bandwidth of less than 16%.

1.02 s with 4K data points, giving a spectral width of 2 kHz. The broad background signal of the liquid crystal was reduced by digital filtering of the audio signal. Figure 3 shows the original deuterium-coupled proton spectra of pentadeuterobenzene at 290 and 300 K, and in Fig. 4 the corresponding deuterium spectra with a p /2 pulse excitation are given. The relevant heteronuclear dipolar coupling constants can be estimated from the proton spectra. At 290 K, the largest coupling constant to the two deuterons in 290 K ortho position to the proton is d DH,ortho /2p É 115 Hz, while at 300 K it decreases, due to motional averaging, to 300 K /2p É 100 Hz. The effective quadrupolar coupling d DH,ortho constants can be determined from the separation of the two groups of lines in the deuterium spectra in Fig. 4 to be vQ/ 2p Å 9.4 kHz at 290 K, and vQ/2p Å 8.1 kHz at 300 K. Comparison of the experimental spectra using the 13 optimized decoupling sequences showed that the calculated performance for an idealized two-spin system strictly correlates with the performance for the six-spin system of pentadeuterobenzene. The pulse sequence TL36 shows the least linewidth, the maximum intensity, and the maximum flatness with respect to the deuterium irradiation offset. TL36 is be compared to CW and COMARO-4 decoupling in Fig. 5 at 290 K. The proton resonance lineshape is shown as a function of the deuterium irradiation offset in increments of 50 Hz. No digital filter functions were applied. The amplitude values are on the same scale and are directly comparable. The linewidths at half-height for zero deuterium irradiation offset are 1.7, 2, and 1.25 Hz for CW, COMARO-4, and TL36 decoupling, respectively. Figure 6 shows analogous spectra taken at 300 K. The linewidths for zero deuterium

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irradiation offset are here 0.9 Hz for the CW, 1.3 Hz for COMARO-4, and 0.8 Hz for the TL36 decoupling. Within the entire deuterium-chemical-shift range of 12 ppm (737 Hz at 400 MHz proton resonance frequency), the performance of TL36 is better than the optimum performance of COMARO-4, which is obtained at an offset of 400 Hz (see Fig. 5). Within a range of 170 Hz (at 290 K) and 230 Hz (at 300 K), it is also better than the behavior of CW decoupling at the exact deuterium-chemical-shift position. With the optimized TL36 sequence, there is still a significant variation of the proton peak amplitudes over the deuteriumdecoupling offset range. The variation is, however, not reflected in the center-peak integrals. At both temperatures, the integrals computed within a frequency range of {4 Hz about the peak maxima vary by less than 0.5% for the whole range of {1 kHz deuterium resonance offset. The analogous variation for COMARO-4-decoupled spectra (with an integration range of {25 Hz) exceeds 15% of the maximum value, indicating some intensity loss into low-amplitude sidebands. When a frequency-offset-independent behavior for the peak amplitude is desired, it is possible to intentionally shift the deuterium irradiation frequency by, e.g., 500 Hz away from the center, making use of the flat amplitude characteristics of TL36 for deuterium frequency offsets larger than 300 Hz. As can be seen from Fig. 7, displaying the performance for large offsets at 290 K, the peak maxima vary within a bandwidth of less than 16% of the maximum value in the offset range from 300 to 1000 Hz. However, the decoupling efficiency, measured by the peak amplitude, is thereby reduced by a factor of 5.5 in comparison to the on-resonance behavior.

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CONCLUSIONS

2. A. J. Shaka, J. Keeler, T. Frenkiel, and R. Freeman, J. Magn. Reson. 52, 335 (1983).

It has been shown in this paper that the derivatives required in the application of an analytical gradient-search procedure can be computed conveniently for arbitrarily complex system Hamiltonians and for arbitrary pulse sequences. This allows the application of gradient searches even in spaces of high dimensionality for quantum-mechanical pulse-sequence optimization tasks. The procedure described here is general. It can be applied for arbitrary pulse-sequence optimizations, provided that the quality function can be expressed in terms of the total propagator and its eigenvalues and eigenvectors. The inherent difficulties of searching for the global minimum on a multimodal error surface, however, are not eliminated by an analytical gradient method, and the possibility of trapping in local minima cannot be avoided. The application of the proposed procedure for the design of optimized deuterium decoupling demonstrates its practical value and feasibility.

3. A. J. Shaka, P. B. Barker, and R. Freeman, J. Magn. Reson. 64, 547 (1985).

We thank Professor W. Gander for stimulating discussions. We are grateful to Christoph Scheurer and Dr. Thomas Schulte-Herbru¨ggen for drawing our attention to Ref. (17). This research has been supported by the Swiss National Science Foundation.

REFERENCES

/

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maga

AP: Mag Res, Series A