Evolution of a spin system under the heteronuclear scalar coupling Hamiltonian during an adiabatic sweep has been treated theoretically by Kay and coworkers (

Zwahlen, et al., 1997,

Zwahlen, et al., 1998). The adiabatic sweep is divided into a series of equal time intervals short enough that the amplitude and resonance offset of the rf field can be treated as constant during each interval. The evolution of the density matrix during the

*i*th interval is calculated using as initial conditions the state of the density operator after the (

*i* − 1)th interval. Thus, evolution through the entire adiabatic sweep is calculated iteratively for the

*N* intervals Δ

*t*_{i} for

*i* = 1,…,

*N*, beginning with the initial conditions at the start of the adiabatic sweep. To simplify these calculations, Kay and coworkers developed compact expressions for efficiently calculating the evolution of a reduced set of basis operators during each interval.

In most applications, and in particular for isotope filtration, the state of the density operator does not need to be known at any arbitrary point within an adiabatic sweep; rather the net evolution of the density operator at the end of the sweep is of primary concern. In these circumstances, an average Hamiltonian theoretical treatment of evolution under the heteronuclear scalar coupling Hamiltonian is sufficiently accurate and yields simple intuitive results. Average Hamiltonian theory has been applied to adiabatic pulses for isolated spin systems, i.e. in the absence of scalar couplings, by Mitschang and Rinneberg (

Mitschang and Rinneberg, 2003) and to homonuclear scalar coupled systems by Bennett and coworkers (

Bennett, et al., 2003).

The experimental situation of interest is illustrated in . The initial state of the density operator for an *I*-*S* spin system (*I* = ^{1}H and *S* = ^{13}C) is described by an expansion in terms of in-phase (*I*_{x} and *I*_{y}) and antiphase transverse *I* spin product operators (2*I*_{x}S_{z} and 2*I*_{y}S_{z}). An adiabatic sweep of duration *τ*_{p} is applied to the *S* spin and a refocusing 180° is optionally applied to the *I* spin at time *τ*_{inv}. The sequence element in is formally equivalent to the sequence -*τ*_{p}-[180°(*I*)] in which the Hamiltonian in the rotating frames of reference of the *I* and *S* spins during the period *τ*_{p} is given by

and the bracketed 180°(*I*) pulse is included only if a refocusing pulse is applied. The refocusing pulse inverts the scalar coupling interaction at time *τ*_{inv}; the effect of this pulse is represented by the function *a*(*t*) with values

Without loss of generality, the

*I* spin is assumed to be on-resonance in the rotating frame. The rotating frame of the

*S* spin is defined relative to the instantaneous radiofrequency field of the adiabatic sweep,

*ω*_{rf}(

*t*); therefore, the frequency offset of the

*S* spin is given by the time-dependent function Ω(

*t*) =

*ω*_{S} −

*ω*_{rf}(

*t*) and

*ω*_{S} is the resonance frequency of the

*S* spin. The amplitude of the adiabatic sweep, assumed to be applied with

*x*-phase is given by

*ω*_{1}(

*t*). The amplitude and the tilt angle of the effective field in the rotating frame are given by

*ω*_{e}(

*t*) = [Ω(

*t*)

^{2}+

*ω*_{1}(

*t*)

^{2}]

^{1/2} and tan[

*θ*(

*t*)] =

*ω*_{1}(

*t*)/Ω(

*t*), respectively. The time-variation of

*ω*_{e}(

*t*) and

*θ*(

*t*) are assumed to satisfy the adiabatic condition, in which the adiabaticity factor

*Q* =

*ω*_{e}/|

*dθ*(

*t*)/

*dt*|

1; typically, the adiabaticity factor at resonance, Ω(

*t*) = 0, is chosen to satisfy Q

_{0} > 5.

Evolution under this Hamiltonian is analyzed most conveniently in a tilted frame of reference whose

*z′*-axis is oriented along the instantaneous direction of

*ω*_{e}(

*t*). Transformation to the tilted frame is defined by

with

*U*_{1} = exp[

*iθ*(

*t*)

*S*_{y}]:

in which

*T* is the Dyson time-ordering operator (

Evans and Powles, 1967), and

is the effective Hamiltonian in an interaction frame rotating around the

*z′*-axis with frequency

*ω*_{e}(

*t*). In this frame, illustrated in , the Hamiltonian is given by:

The average Hamiltonian is

in which

In the applications envisioned, the transverse components of the interaction frame Hamiltonian given in

Eq. [7], are negligible owing to rapid evolution of the sinusoidal terms that depend on

*ω*_{e}(

*t*). Thus, the zero-order average Hamiltonian is given by

in which the reduced scalar coupling constant is defined by

In the absence of a refocusing pulse:

in which angle brackets indicate the average value over the adiabatic sweep. The density operator in the laboratory frame at the end of the adiabatic sweep is given by:

The second equality is obtained because the propagator

**U**_{0}(

*τ*_{p}) commutes with

and the initial density operator and because the adiabatic sweep achieves a total rotation

*θ*(

*τ*_{p}) =

*π*. Thus, evolution of the density operator during

*τ*_{p} is obtained by the simple product operator rules for evolution under a reduced scalar coupling Hamiltonian

followed by an ideal

*πS*_{y} rotation; for example:

If a 180°(*I*) refocusing pulse is applied during the adiabatic sweep, then these results are modified by applying the 180°(*I*) pulse to the operators present after *τ*_{p}.

For a chirp pulse, a linear frequency sweep with a constant value of

*ω*_{1},

Eq. [13] can be integrated to give

in which ΔF is the sweep bandwidth and

*δ* is the frequency offset between the

*S* spin and the center of the frequency sweep. The second line is obtained assuming that

. When this approximation is valid, evolution of the scalar coupling during an adiabatic sweep (in the absence of a refocusing 180° pulse applied to the

*I* spins), can be represented as the sequence element

*τ′* − 180°(

*S*) − (

*τ*_{p} −

*τ′*), in which

In

Eq. [17],

*τ′* is equal to the time at which the adiabatic frequency sweep is resonant with the

*S* spin (

Zwahlen, et al., 1997). These results are approximately correct for the WURST family of adiabatic pulses (

Kupče and Freeman, 1995) as well, provided that

*δ* is restricted to the central constant-amplitude region of the pulse.

The accuracy of the average Hamiltonian result of

Eq. [12] is illustrated in for the pulse sequence element of using a series of WURST-20 adiabatic pulses (

Kupče and Freeman, 1995) with increasing field strengths and shorter durations, typical of values used for these pulses in biological NMR spectroscopy. As shown in , the accuracy of the average Hamiltonian result is weakly dependent on the adiabaticity of the sweep, provided that adiabaticity is maintained (

*Q*_{0} ≥5). However, accuracy depends strongly on

*ω*_{1}_{max}τ_{p}, in which

*ω*_{1}_{max} is the peak value of

*ω*_{1}(

*t*), for two reasons. First, the accuracy of the approximation

, that is, keeping only the zero-order Hamiltonian in

Eq. [8], requires that

*ω*_{1}_{max} *πJ*_{IS}. Second, the cosine and sine terms that depend on

*ω*_{e}(

*t*)

*t* in

Eq. [7] are averaged to zero only if

*ω*_{1}_{max}τ_{p} 1. Thus, the accuracy of the above results is reduced for either weak rf fields or short pulse lengths.

Rotations under on-resonance radiofrequency pulses in the operator space {

*I*_{x},

*I*_{y},

*I*_{z}} are isomorphous to rotations under the scalar coupling Hamiltonian in the operator space {2

*I*_{z}S_{z}, 2

*I*_{x}S_{z},

*I*_{y}} (

Levitt, 1982). Thus, a composite 90° pulse that efficiently rotates

*I*_{z} magnetization into the transverse plane independently of variation in the nominal rotational angle can be converted into a pulse-interrupted-free-precession period that transforms

*I*_{y} magnetization efficiently into two-spin operators independently of variation in

*J*_{IS}. The composite 90° pulse sandwich

*β*_{0}(2

*β*)

_{2}_{π}_{/3}, in which the bracketed term is the nominal rotation angle and the subscript is the phase of the pulse, has previously been used by Stuart and coworkers to develop a highly efficient composite-rotation isotope filter (

Stuart, et al., 1999). Using the aforementioned isomorphism, the composite filter propagator is given by:

The leftmost *I*_{y} rotation is not needed to implement an isotope filter and can be eliminated to simplify the propagator to

in which *β* = *πJτ* and the filter is tuned to for a nominal scalar coupling constant *J*_{0} by setting the evolution delay *τ* = 1/(2*J*_{0}). This propagator is realized by the ideal filter pulse sequence:

A product operator analysis of the sequence shows that the evolution of *I*-spin coherence between points *a* and *b* of is given by −*I*_{y} → −*εI*_{y} in which:

For a filter, the two-spin product operators created in this sequence, 2*I*_{z}S_{z} and 2*I*_{x}S_{z}, are purged by a 90° pulse applied to the *S* spins. In other applications, optimal transfer to 2*I*_{z}S_{z} longitudinal two-spin order can be obtained by appending a final 30°_{−}_{y} rotation to the sequence. For ^{1}H spins attached to ^{12}C nuclei, *J*_{IS} = 0 and *ε* = 1 under all conditions (ignoring relaxation).

As emphasized by Kay and coworkers (

Zwahlen, et al., 1997) and by Kupče and Freeman (

Kupče and Freeman, 1997), one-bond C-H scalar coupling constants and isotropic chemical shifts are linearly related for

^{1}H spins attached to

^{13}C nuclei in amino acid residues and RNA nucleotides:

in which

*J*_{IS} is measured in Hz and

*δ*_{C} is the chemical shift measured in ppm (relative to DSS). As shown by

Eq. [12], the reduced scalar coupling constant during an adiabatic sweep also depends on the

*S*-spin resonance offset. Accordingly, the pulse element shown in can be substituted for the central pair of 180° pulses in an INEPT element. The properties of the adiabatic sweep are chosen to obtain broadband polarization transfer from initial −

*I*_{y} magnetization to 2

*I*_{x}S_{z} for the desired range of

*J*_{IS}. This strategy has been elegantly incorporated into double-tuned isotope filters by Kay and coworkers (

Zwahlen, et al., 1997).

Two implementations of a joint composite-rotation adiabatic-sweep filter element are shown in . The two filter sequences have essentially the same theoretical performance. In the sequence of , the two adiabatic pulses have the same frequency bandwidth and are swept in the same sense; therefore, the values of

_{IS} calculated using

Eq. [12] are identical for the two adiabatic sweeps. A product operator analysis of the sequence shows that the evolution of

*I*-spin coherence between points

*a* and

*b* of is given by −

*I*_{y} → −

*εI*_{y} in which:

In the sequence of , the second adiabatic pulse, of length

*τ*_{p}, has the same sweep bandwidth, but is swept in the opposite sense as the first adiabatic sweep. Therefore <cos[

*θ*(

*t*)]>

_{sweep2} = −<cos[

*θ*(

*t*)]>

_{sweep1} = <cos

*θ*> using

Eq. [13]. A product operator analysis of the sequence shows that the evolution of

*I*-spin coherence between points

*a* and

*b* of is given by −

*I*_{y} → −

*εI*_{y} in which:

The filters are implemented by numerically optimizing the adiabatic sweeps in order to minimize

*ε* in

Eqs. [24] and

[25] over the desired range of

*J*_{IS} (while maintaining adiabaticity of the sweep as a constraint). For

^{1}H spins attached to

^{12}C nuclei,

*J*_{IS} = 0 and

*ε* = 1 under all conditions (ignoring relaxation). compares the theoretical performance of the joint composite-rotation adiabatic-sweep isotope filter for scalar coupling constants in proteins with other approaches that have been described in the literature. The corresponding theoretical performance for scalar coupling constants in RNA is shown in . The improvements obtained with the new sequences, particularly at the low and high extremes of the scalar coupling constants, are evident. The sequence of is derived from the basic composite rotation

*β*_{0}(2

*β*)

_{2}_{π}_{/3} in a more straightforward fashion; however, as discussed below, the sequence of has only 180° pulses on the

*I*-spin when transverse coherences are present and allows convenient water suppression with the excitation sculpting (

Hwang and Shaka, 1995) approach. In addition, resonance offset effects are smaller for a 60° pulse compared with a 120° pulse.

The above theoretical analyses do not include the effects of relaxation. The joint composite-rotation adiabatic-sweep isotope filters are longer than the corresponding composite-rotation sequences of Stuart and coworkers (

Stuart, et al., 1999), for example, the sequence of is longer by 3

*τ*_{p}/2, leading to reduced sensitivity, owing to relaxation losses. Thus, the lengths of the adiabatic sweeps also should be minimized while satisfying

Eqs. [24] and

[25]. Transverse relaxation for

^{1}H spins attached to

^{13}C nuclei is more efficient than for

^{1}H spins attached to

^{12}C nuclei, because the

^{1}H-

^{13}C dipolar interaction constitutes an additional relaxation pathway. Consequently, the apparent filter efficiency, measured as the ratio of peak intensities for

^{1}H spins attached to

^{12}C nuclei and for

^{1}H spins attached to

^{13}C nuclei is increased in larger proteins with larger transverse relaxation rate constants.