# Continuous Floquet Theory in Solid-State NMR

Matías Chávez  Matthias Ernst<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of
Chemistry and Applied Biosciences, ETH Zürich, Vladimir-Prelog-Weg 2, 8093
Zürich, Switzerland

###### Abstract

This article presents the application of continuous Floquet theory in solid-
state NMR. Continuous Floquet theory extends traditional Floquet theory to
non-continuous Hamiltonians, enabling the description of observable effects
not fully captured by traditional Floquet theory due to its requirement for a
periodic Hamiltonian. We present closed-form expressions for computing first
and second-order effective Hamiltonians, streamlining integration with
traditional Floquet theory and facilitating application in NMR experiments
featuring multiple modulation frequencies. Subsequently, we show examples of
the practical application of Continuous Floquet theory by investigating
several solid-state NMR experiments. These examples illustrate the importance
of the duration of the pulse scheme regarding the width of the resonance
conditions and the near-resonance behavior.

††preprint: AIP/123-QED

##  I Introduction

The ability to analyze the time evolution of the density operator under time-
dependent Hamiltonians is crucial for understanding and designing experiments
in solid-state NMR Haeberlen (1976); Mehring (1983); Schmidt-Rohr and Spiess
(1994). In solid-state NMR, time-dependent Hamiltonians arise from mechanical
sample rotation and the application of radio-frequency (rf) irradiation.
Sample-rotation techniques like magic-angle spinning (MAS) Andrew, Bradbury,
and Eades (1958); Lowe (1959), dynamic-angle spinning (DAS)Mueller  _et al._
(1990), or double rotation (DOR) Samoson, Lippmaa, and Pines (1988) are
employed to average anisotropic interactions and achieve high-resolution
spectra. Radio-frequency irradiation is used to manipulate the spin part of
the Hamiltonian, either further eliminating or reintroducing certain terms
that are averaged by sample rotation.

In solid-state NMR under MAS, multiple time dependencies are common and play a
significant role in many experiments. In the simplest case of MAS without any
additional rf irradiation, only a single frequency appears in the Hamiltonian.
However, even without rf irradiation, multiple frequencies might be required
for example to describe the rotational-resonance (R2) experiment Colombo,
Meier, and Ernst (1988); Raleigh, Levitt, and Griffin (1988); Levitt  _et al._
(1990) where a chemical-shift interaction frame is often used to avoid
combined transitions in spin and Fourier space.

In many experiments involving sample rotation and rf irradiation on a single
channel, at least two frequencies arise that can be commensurate (rotor-
synchronized rf irradiation) or incommensurate. For cyclic pulse sequences,
i.e., a sequence that has no net rotation, a single frequency is sufficient to
describe the interaction-frame transformation. Examples of such experiments
that require two frequencies include continuous-wave (cw) irradiation, phase-
modulated Lee–Goldburg (PMLG) decoupling Vinogradov, Madhu, and Vega (1999,
2000), symmetry-based recoupling sequences Levitt (2007), or XiX irradiation
Tekely, Palmas, and Canet (1994); Detken  _et al._ (2002) for heteronuclear
decoupling.

Experiments with three independent frequencies include MAS with constant
irradiation on two different spin species. There are experiments involving MAS
with a constant rf irradiation on two different spins, like in Hartmann-Hahn
cross-polarization (CP) Hartmann and Hahn (1962); Pines, Gibby, and Waugh
(1973); Stejskal, Schaefer, and Waugh (1977) or a combination of constant rf
irradiation with a cyclic multiple-pulse sequence as in the resonant second-
order recoupling (RESORT) experiment Scholz, Meier, and Ernst (2010). Pulse
sequences that are not cyclic, i.e., sequences that have a net rotation,
always require two frequencies to characterize the interaction frame. Such
sequences require three frequencies in combination with MAS, such as two-pulse
phase-modulated (TPPM) decoupling Bennett  _et al._ (1995); Scholz  _et al._
(2009). If pulse imperfections like rf-field inhomogeneity, pulse transients
or chemical-shift offsets are taken into account in the interaction-frame
transformation, even cyclic sequences acquire a net rotation and require two
independent frequencies. Experiments involving cw irradiation and a chemical-
shift interaction-frame transformation, such as the mixed rotational and
rotary-resonance recoupling (MIRROR) sequence Scholz  _et al._ (2008); Scholz,
Meier, and Ernst (2009); Wittmann  _et al._ (2014); Chávez and Ernst (2022a),
require also three frequencies. Additionally, if the influence of the
isotropic chemical shift is considered, the number of independent frequencies
increases and examples with up to five frequencies can be found in the
literature Shankar  _et al._ (2017).

Various approaches can be employed to handle time-dependent Hamiltonians and
derive a propagator for the entire sequence. The most commonly used method in
NMR is average-Hamiltonian theory (AHT) Haeberlen (1976); Mehring (1983),
based on the Magnus expansion. AHT approximates the periodic time-dependent
Hamiltonian by a series of time-independent average Hamiltonians over integer
multiples of the cycle time. However, AHT has limitations; it requires
defining a single basic frequency and cycle time for the Hamiltonian, and the
time evolution of the density operator is accurately described only at
multiples of the cycle time. Consequently, when dealing with multiple
incommensurate time-dependent processes in solid-state NMR, such as sample
rotation and non-synchronized rf irradiation, AHT is often impractical unless
the time scales of the two processes are well-separated and sequential
averaging can be applied.

In contrast, Floquet theory Shirley (1965); Vega (2007); Scholz, Beek, and
Ernst (2010); Leskes, Madhu, and Vega (2010); Ivanov  _et al._ (2021) offers a
more comprehensive approach to describe the time evolution of the spin system
at all times and can handle multiple incommensurate frequencies. Nonetheless,
the matrix representation of the full Floquet Hamiltonian is infinite-
dimensional and can be challenging to intuitively comprehend its implications
on the time evolution of the spin system. An operator-based formulation of the
Floquet Hamiltonian, introduced by Boender Boender (1996) and Augustine
Augustine (1995), presents an opportunity to employ analytical block-
diagonalization techniques like the van Vleck transformation Van Vleck (1948);
Primas (1961, 1963); Goldman  _et al._ (1992). This leads to an approximately
diagonalized Floquet Hamiltonian Vinogradov, Madhu, and Vega (2001); Ramesh
and Krishnan (2001); Ernst, Samoson, and Meier (2005), which can then be
projected back into Hilbert space to obtain an effective time-independent
Hilbert-space Hamiltonian. This effective Hamiltonian allows for a more
intuitive discussion of the time evolution of the spin system Ernst, Samoson,
and Meier (2005).

Multi-mode operator-based Floquet theory becomes particularly important for
dealing with non-resonant problems, which can be challenging to analyze using
average-Hamiltonian theory (AHT). Non-resonant experiments, such as
homonuclear Vinogradov, Madhu, and Vega (2004a) or heteronuclear Madhu (2013)
decoupling using continuous-wave (cw) irradiation or multiple-pulse sequences,
fall into this category. Using Floquet theory to derive effective
Hamiltonians, the analysis of experiments involving resonant and non-resonant
second-order effects becomes more straightforward. Such second-order
polarization-transfer schemes are less susceptible to dipolar-truncation
effects Hohwy  _et al._ (1999a, b); Grommek, Meier, and Ernst (2006); Bayro
_et al._ (2009) compared to first-order recoupling sequences and are used to
obtain long-range distance restraints in proteins Manolikas, Herrmann, and
Meier (2008); De Paëpe  _et al._ (2008), making them valuable for applications
in biological solid-state NMR.

However, an important limitation of Floquet theory lies in its requirement for
a periodic Hamiltonian. Consequently, Floquet theory always describes the
periodic continuation of the actual Hamiltonian, implying a continuous
repetition of the experiment. In the course of this paper, it will become
evident that certain observable effects are not fully captured by traditional
Floquet theory due to this mandatory periodic continuation. Notably, effects
such as the width of resonance conditions, recoupling and decoupling
performance under MAS detuning, and chemical-shift selective recoupling
Potnuru  _et al._ (2021); Xiao, Zhang, and Yang (2021); Zhang  _et al._ (2019,
2020); Chávez and Ernst (2022b) are often difficult to describe using Floquet
theory. Another shortcoming of Floquet theory (and AHT) is the inability to
adequately describe near-resonance conditions due to the slow convergence of
the non-resonant series expansion. However, such descriptions are important
for the numerical optimization of effective Hamiltonians. One possible
workaround is the use of the nearest resonance condition and adding the
detuning from the resonance condition to the effective Hamiltonian Shankar
_et al._ (2017). However, such an approach is only possible for continuous-
wave and not for amplitude or phase-modulated irradiation.

Continuous Floquet theory Chávez and Ernst (2022a) offers a more complete
description by eliminating the requirement of a periodic continuation of the
Hamiltonian. This framework describes the original Hamiltonian itself, rather
than its periodic continuation, introducing the duration of the Hamiltonian as
a parameter. The finite and duration-dependent resonance conditions that arise
provide a comprehensive explanation and quantification of various effects.
These include performance deterioration resulting from magic-angle spinning
(MAS) or rf-field amplitude mismatch, as well as the impact of chemical-shift
offset and incomplete averaging Chávez and Ernst (2022b).

Nonetheless, the theory outlined in Chávez and Ernst (2022a) does have a
limitation: it amalgamates distinct modulations, such as sample rotation, rf-
field irradiation, and chemical shift, into a singular basic frequency, unlike
traditional Floquet theory, which explicitly segregates these modulations
using distinct basic frequencies. Additionally, the effective Hamiltonians of
varying orders are not readily available in closed forms; their computation
entails the evaluation of integrals, rendering their application less
convenient.

In the first part of this paper, we tackle these limitations by developing
closed-form expressions for the effective Hamiltonians, without the
introduction of assumptions or approximations. Furthermore, we establish an
explicit correlation between the various modulations using multiple basic
frequencies, akin to the effective Hamiltonians found in the traditional
Floquet theory.

In the second part of the paper, we apply the derived effective Hamiltonian to
a range of solid-state NMR experiments, spanning from rotary resonance
experiments and extending to multi-pulse recoupling sequences such as
symmetry-based $\mathrm{C}$\- and $\mathrm{R}$-type pulse schemes. Within the
context of continuous Floquet theory, we investigate how the width of the
resonance conditions is influenced by the duration of the pulse schemes,
providing a comprehensive understanding of their behavior in practical
applications.

##  II Theory or Effective Hamiltonians in Continuous Floquet Theory

The first and second-order terms of the effective Hamiltonian in continuous
Floquet theory are given as Chávez and Ernst (2022a)

$\displaystyle\bar{{\mathcal{H}}}^{(1)}$
$\displaystyle=\frac{1}{T}\widehat{{\mathcal{H}}}(\Omega=0)$ (1)
$\displaystyle\bar{{\mathcal{H}}}^{(2)}$
$\displaystyle=-\frac{1}{2\,T}\,PV\int\frac{[\widehat{{\mathcal{H}}}(\Omega),\widehat{{\mathcal{H}}}(-\Omega)]}{\Omega}\textrm{d}\Omega~{}.$
(2)

The quantity $\widehat{{\mathcal{H}}}(\Omega)$ is the Fourier transform of the
Hamiltonian defined as

$\displaystyle\widehat{{\mathcal{H}}}(\Omega)$
$\displaystyle=\int\limits_{-\infty}^{\infty}{\mathcal{H}}(t)e^{-i\Omega
t}\textrm{d}t~{},$ (3) $\displaystyle{\mathcal{H}}(t)$
$\displaystyle=\frac{1}{2\pi}\int\limits_{-\infty}^{\infty}\widehat{{\mathcal{H}}}(\Omega)e^{i\Omega
t}\textrm{d}\Omega~{},$ (4)

where ${\mathcal{H}}(t)$ is the time-dependent Hamiltonian of the system under
investigation and $T$ is the total duration of this Hamiltonian. Note that the
normalization of the Fourier transform has been changed compared to the
original publication of continuous Floquet theory Chávez and Ernst (2022a) to
obtain a simpler agreement with traditional Floquet theory in the case of
periodic continuation of a sequence. These formal definitions of the effective
Hamiltonians are not explicitly dependent on experimental parameters that
generate the time dependence of the Hamiltonian. A single frequency
characterizes multiple time-dependent processes generated, for example, by the
sample rotation , interaction-frame transformations to describe the rf-field
irradiation or the chemical-shift offset
The first-order effective Hamiltonian of Eq. (1) is fairly straightforward
involving only the evaluation of the Fourier transform of the Hamiltonian at
frequency $\Omega=0$. The second-order effective Hamiltonian of Eq. (2), is
given as a Cauchy principal value Kanwal (1997), which has to be evaluated to
obtain a useful result. This can be limiting, especially if we want to design
computationally efficient tools because the principal value has to be solved
numerically every time we change the experimental parameters. In addition,
solving principal values can be challenging, and in general, require special
treatments near the pole of the integral kernel. These treatments often imply
higher sampling rates of the numerical integration algorithm and, thus,
increases the computational cost significantly. A closed-form expression of
effective Hamiltonian, therefore, increases interpretability and significantly
reduces the computational cost. Hence enabling a more efficient and intuitive
approach for the application of continuous Floquet theory without impeding its
generality.
The effective Hamiltonians at various orders do not rely on any assumptions or
approximations. As a consequence, the duration of the Hamiltonian $T$ is
introduced in the effective theory, which is missing in traditional Floquet
theory, because we assume that the Hamiltonian is periodic. In other words, we
approximate the ’real’ Hamiltonian with a Hamiltonian, which is periodically
continued.
In the same way, any Hamiltonian with a finite duration can be expressed as a
section of a periodic Hamiltonian. In the following, we are going to explore
this idea of periodic continuation and the application of a time-window
function in combination with continuous Floquet theory. As a result, we will
obtain simpler and more expressive closed-form expressions for the effective
Hamiltonians.

###  II.1 Periodic Continuation and Time Window Function

Every Hamiltonian of a finite duration can be seen as a section of a
Hamiltonian with a longer duration. Thus we can formulate a Hamiltonian as a
product of a continuation of itself and a rectangular function representing a
time window. At first, this might seem redundant, but allows for substantial
simplification of the expressions of the effective Hamiltonians as shown in
the following. Let us consider a time-dependent Hamiltonian that spans the
interval $T$ from $t_{0}=-T/2$ to $t_{\mathrm{f}}=+T/2$. In principle, any
arbitrary time interval of size $T$ can be chosen, but this choice is
symmetric around zero, which will turn out to be convenient later because the
Fourier transform of a symmetric function is real. That said, a non-symmetric
choice of time interval is possible but will lead to slightly different and,
arguably, more complicated expressions for the final effective Hamiltonian.
However, the steps of the derivation are in both cases the same and no
additional difficulties arise from the choice of symmetry given by the time
interval. As mentioned before, the main idea is to represent the Hamiltonian
as a product of its periodic continuation to the entire time domain and a
rectangular window function $\Pi(t/T)$ that ’cuts out’ the desired interval
$T$:

${\mathcal{H}}(t)=\widetilde{{\mathcal{H}}}(t)\,\Pi(t/T)~{}.$ (5)

$\widetilde{{\mathcal{H}}}(t)$ denotes a periodic continuation of the
Hamiltonian to the entire time domain ($t\in\mathbb{R}$) and $\Pi(t/T)$ is a
rectangular window function

$\Pi(t/T)=\begin{dcases}1&|t|<T/2\\\ 0&\mathrm{else}\end{dcases}~{}.$ (6)

Eq. (5) is a reformulation of the Hamiltonian and does not introduce any
assumption or restrictions on the Hamiltonian. A graphical illustration of Eq.
(5) is given in Fig. 1. Note, that in traditional Floquet theory only the
periodic continuation $\widetilde{{\mathcal{H}}}(t)$ is considered and not the
original Hamiltonian.

Figure 1: Illustration of the reformulation of a time-dependent Hamiltonian in
terms of a periodic continuation of the Hamiltonian
$\widetilde{{\mathcal{H}}}$ and a rectangular window function. In this example
the Hamiltonian is periodically continued with the period $T$, i.e. with the
frequency $\omega_{\mathrm{p}}=2\pi/T$.

In addition, we would like to mention, that any continuation
$\widetilde{{\mathcal{H}}}(t)$ satisfies Eq. (5), because the window function
$\Pi(t/T)$ is only non-zero in the time-interval $t\in[-T/2,T/2]$ of the
original Hamiltonian ${\mathcal{H}}(t)$. Hereafter, we consider only periodic
continuations of the Hamiltonian, where the Hamiltonian is extended to the
entire real domain periodically. A main advantage of using periodic
continuations is, that we can formulate it in terms of a Fourier series: any
periodic continued Hamiltonian can be written as a Fourier series with a basic
frequency $\omega_{\mathrm{p}}=2\pi/T$:

$\widetilde{{\mathcal{H}}}(t)=\sum_{n}\widetilde{{\mathcal{H}}}^{(n)}e^{in\omega_{\mathrm{p}}t}$
(7)

with

$\displaystyle\widetilde{{\mathcal{H}}}^{(n)}$
$\displaystyle=\frac{1}{T}\int_{-T/2}^{T/2}{\mathcal{H}}(t)e^{-in\omega_{\mathrm{p}}t}\textrm{d}t~{}.$
(8)

Insertion into Eq. (5), immediately leads to the Hamiltonian

${\mathcal{H}}(t)=\widetilde{{\mathcal{H}}}(t)\Pi(t/T)=\sum_{n}\widetilde{{\mathcal{H}}}^{(n)}e^{in\omega_{\mathrm{p}}t}\,\Pi(t/T)~{}.$
(9)

Here we considered the simplest periodic continuation, where the Hamiltonian
is extended to the entire time domain by just repeating the original
Hamiltonian, depicted schematically in Fig. 2 a). The period of this periodic
continuation is the overall duration of the original Hamiltonian $T$, and
hence the basic frequency is $\omega_{\mathrm{p}}=2\pi/T$. Any Hamiltonian can
be represented using Eq. (9) with the basic frequency $\omega_{\mathrm{p}}$.
While this is simple to implement, it might require high-order Fourier
coefficients in $\omega_{\mathrm{p}}$ because the frequency is not matched to
the frequencies of the Hamiltonian and discontinuities are present in the
periodic continuation of the Hamiltonian.

If the basic frequencies and the functional form of the Hamiltonian are known,
as it is often the case in solid-state NMR, then the periodicity of the
Hamiltonian ($\tau_{\mathrm{r}}$) is known and we can also use a periodic
continuation of the Hamiltonian based on the basic frequencies of the
Hamiltonian. Fig. 2 b) shows an example of a different periodic continuation
which has the period $\tau_{\mathrm{r}}<T$. This periodic continuation also
reproduces the original Hamiltonian in the relevant interval $t\in[-T/2,T/2]$
and can be used in Eq. (9). Such a periodic continuation might be more complex
to implement but will require fewer Fourier coefficients to characterize the
time-dependent Hamiltonian by avoiding discontinuity. It is important to
notice, that a different periodic continuation will always have a different
set of Fourier coefficients and Fourier frequencies according to Eq. (8).

If the Hamiltonian is modulated by several independent processes like MAS and
rf irradiation characterized by an interaction-frame transformation, multiple
basic frequencies are present and it might be best to define the periodic
continuation using a multi-dimensional Fourier series on the set of basic
frequencies. Further, note that a continuous periodic continuation of
$\widetilde{\mathcal{H}}$ is not required. Therefore, the discontinuities of
the periodic continuation in Fig. 2 a) do not cause any problems but are
rather an inconvenience giving rise to higher-order Fourier coefficients than
in the continuous case (Fig. 2 b)). This could increase the computational cost
especially when calculating second- or higher-order terms of the effective
Hamiltonian. This is because the periodic continuation of the Hamiltonian in
continuous Floquet theory is only used to reformulate the original Hamiltonian
using Eq. (9). Any periodic extension is applicable; the choice of which one
to utilize primarily depends on convenience. In traditional Floquet theory,
the situation is different. The periodic continuation of the Hamiltonian
serves as an approximation of the original Hamiltonian, i.e., it is the
Hamiltonian that we describe in traditional Floquet theory. Therefore we have
to ensure that the periodic continuation is a good approximation, which, for
example, excludes periodic continuations that are not continuous.

Figure 2: A simplified illustration of two different periodic continuations of
the same Hamiltonian. (a) Periodic continuation, where the Hamiltonian is
simply repeated. (b) Periodic continuation with the period $\tau_{\mathrm{r}}$
of the Hamiltonian that is shorter than the time interval $T$. Both periodic
continuations are equal in the relevant interval $t\in[-T/2,T/2]$. The arising
discontinuities in (a) are unproblematic in continuous Floquet theory since we
only consider the interval $t\in[-T/2,T/2]$ but will lead to artifacts in
traditional Floquet theory since the description outside the relevant interval
is not correct.

Two primary factors cause the time-dependence of the Hamiltonians in solid-
state NMR: spatial rotation of the sample about one or more axes, and
rotations in the spin space caused by rf-field irradiation that can be
described by interaction-frame transformations. Because rotations of the
spatial and the spin space are independent of each other, the Hamiltonian can
be formulated as a sum scalar products between the spatial parts
$\vec{A}_{l}(t)$ and the spin parts $\vec{\mathcal{I}}_{l}(t)$, i.e.

${\mathcal{H}}(t)=\sum_{l}\vec{A}_{l}(t)\vec{\mathcal{I}}_{l}(t)~{}.$ (10)

Typically in solid-state NMR the spatial part is modulated due to MAS and can
be written as a Fourier series in the MAS frequency $\omega_{\mathrm{r}}$:

$\vec{A}_{l}(t)=\sum_{n=-l}^{l}\vec{A}_{l}^{(n)}e^{in\omega_{\textrm{r}}t}~{},$
(11)

with

$\vec{A}_{l}^{(n)}=\frac{1}{\tau_{\textrm{r}}}\int_{-\tau_{\textrm{r}}/2}^{\tau_{\textrm{r}}/2}\vec{A}_{l}(t)e^{-in\omega_{\textrm{r}}t}\textrm{d}t~{}.$
(12)

In the usual rotating frame, the time dependence of the spin part
$\vec{\mathcal{I}}_{l}(t)$ is often due to the rf-field irradiation which we
can describe by an interaction-frame transformation. For a general periodic
sequence that is cyclic, i.e., the interaction frame can be described by an
identity propagator over a whole cycle, only a single frequency,
$\omega_{\mathrm{m}}=2\pi/\tau_{\mathrm{m}}$ is required to characterize the
time-dependent interaction-frame Hamiltonian. If the interaction frame is not
cyclic, i.e., if the interaction frame has an effective field nutation over
the cycle time $\tau_{\mathrm{m}}$, employing a Fourier series with two
frequencies instead of a single basic frequency can be advantageous. This is
particularly relevant for multiple-pulse sequences, where the Hamiltonian
exhibits multiple time dependencies, and also when dealing with multiple
chemical shifts. In these scenarios, the use of multiple frequencies in the
Fourier series proves to be more insightful for an accurate periodic
continuation of Hamiltonian. A common choice of frequencies is using the
modulation frequency $\omega_{\text{m}}$ of the multi-pulse sequence and the
effective field frequency $\omega_{\textrm{eff}}=\beta/\tau_{\text{m}}$, where
$\beta$ is the effective flip angle of the periodic sequence.

$\vec{\mathcal{I}}_{l}(t)=\sum_{k}\sum_{\ell=-l}^{l}\vec{\mathcal{I}}^{(k,\ell)}_{l}e^{ik\omega_{\mathrm{m}}t}e^{i\ell\omega_{\textrm{eff}}t}\Pi(t/T)~{},$
(13)

with

$\vec{\mathcal{I}}^{(k,\ell)}_{l}=\frac{1}{T}\int_{-T/2}^{T/2}\vec{\mathcal{I}}_{l}(t)e^{-i(k\omega_{\mathrm{m}}+\ell\omega_{\textrm{eff}})t}\textrm{d}t~{}.$
(14)

Therefore the complete Hamiltonian can be written as

$\displaystyle\widetilde{{\mathcal{H}}}(t)$
$\displaystyle=\sum_{l}\sum_{n,k,\ell}\vec{A}_{l}^{(n)}e^{in\omega_{\textrm{r}}t}\vec{\mathcal{I}}^{(k,\ell)}_{l}e^{i(k\omega_{\text{m}}+\ell\omega_{\textrm{eff}})t}$
$\displaystyle=\sum_{n,k,\ell}\widetilde{{\mathcal{H}}}^{(n,k,\ell)}e^{i(n\omega_{\textrm{r}}+k\omega_{\mathrm{m}}+\ell\omega_{\textrm{eff}})t}$
(15)

The choice of characteristic frequencies to express the time-dependent
Hamiltonian depends on the problem and the interaction frame chosen to
describe the problem. For example, only $\omega_{\textrm{r}}$ and
$\omega_{\text{m}}$ are required, if we have a cyclic interaction frame, that
does not generate an effective field $\omega_{\textrm{eff}}$. For experiments
using different irradiation schemes on multiple spins, multiple interaction
frames might expose additional frequencies that modulate the spin part of the
Hamiltonian. For other sample rotation techniques like double rotation,
multiple spinning frequencies might be required to describe the spatial part
of the Hamiltonian. Alternatively, it is always possible to use only a single
characteristic frequency ($\omega_{\mathrm{p}}=2\pi/T$) or the definition of
the effective Hamiltonians (Eqs. (1) and (2)) directly.

Multi-index notation provides a concise way to write the Hamiltonians,
independent of the chosen periodic continuation and choice of characteristic
frequencies:

${\mathcal{H}}(t)=\widetilde{{\mathcal{H}}}(t)\Pi(t/T)=\sum_{\mathbf{n}}\widetilde{{\mathcal{H}}}^{(\mathbf{n})}e^{i\omega_{\mathbf{n}}t}\Pi(t/T)~{}.$
(16)

Here $\mathbf{n}$ is the multi index, which is a tuple containing all the
indices. For Eq. (II.1) the multi index is given as $\mathbf{n}=(n,k,\ell)$
and
$\omega_{\mathbf{n}}=n\omega_{\mathrm{r}}+k\omega_{\mathrm{m}}+\ell\omega_{\textrm{eff}}$.
It is important to recognize, that $\omega_{\mathbf{n_{0}}}=0$ are the
resonance conditions, where $\mathbf{n}_{0}$ indicate the multi indices that
lead to a resonance condition.
The appearance of specific characteristic frequencies in the effective
Hamiltonian is a matter of choice and different choices are always possible.
This is similar to the choice of coordinate system or interaction frame.
Although many choices are possible, only a few are convenient for a specific
problem. Usually, the choice of a suitable set of characteristic frequencies
is straightforward and arises naturally as will become evident in Section III.

###  II.2 Effective Hamiltonian

Based on the general representation of the time-dependent Hamiltonian given in
Eq. (16) we can simplify the effective Hamiltonians (Eqs. (1) and (2))
significantly. The resulting effective Hamiltonians are simple closed-form
expressions that are generally applicable. Before we can begin to derive the
actual effective Hamiltonian we have to calculate the Fourier transformation
of the time-dependent Hamiltonian to obtain the frequency-space Hamiltonian
$\widehat{{\mathcal{H}}}(\Omega)$. For the (element-wise) Fourier transform
(Eq. (3)) of the Hamiltonian (Eq. (16)), it follows immediately

$\widehat{{\mathcal{H}}}(\Omega)=T\sum_{\mathbf{n}}{\mathcal{H}}^{(\mathbf{n})}\mathrm{sinc}\left(\frac{(\omega_{\mathbf{n}}-\Omega)T}{2}\right)$
(17)

where $\mathrm{sinc}\left(x\right)=\sin(x)/x$. For the sake of clarity, the
derivation is not shown here but can be found in the SI. Note, that we have an
explicit dependence on $T$, $\omega_{\mathbf{n}}$ and $\Omega$ in the factor
$\mathrm{sinc}\left((\omega_{\mathbf{n}}-\Omega)T/2\right)$. The
representation of the Hamiltonian in Eq. (17) is independent of the details of
the frequencies required to describe the Hamiltonian. This is important
because it allows us to obtain general closed-form solutions of the effective
Hamiltonians since we can carry out the integration for $\Omega$ in the
definition of the effective Hamiltonians (Eq. (1) and (2)), as shown in the
following two sections.

####  II.2.1 First-order effective Hamiltonian

The first-order effective Hamiltonian can simply be obtained by inserting the
Fourier Series of (Eq. (17)) in the formal definition of the first-order
effective Hamiltonian Eq. (1), leading to

$\displaystyle\bar{{\mathcal{H}}}^{(1)}=\frac{1}{T}\widehat{{\mathcal{H}}}(0)$
$\displaystyle=\sum_{\mathbf{n}}\widetilde{\mathcal{H}}^{(\mathbf{n})}\,h^{(1)}_{\mathbf{n}}(T)~{},$
(18)

with

$h^{(1)}_{\mathbf{n}}(T)=\text{sinc}\left(\frac{\omega_{\mathbf{n}}T}{2}\right)~{}.$
(19)

Eq. (18) is an expansion of the first-order effective Hamiltonian in terms of
sinc functions $h^{(1)}_{\mathbf{n}}(T)$, similar to the Whittaker–Shannon
interpolation formula Whittaker (1915); Shannon (1949). Fig. 3 shows the
function $h^{(1)}_{\mathbf{n}}(T)$ as function of $\omega_{\mathbf{n}}$ for
different overall duration $T$. The longer the duration $T$ the narrower the
function $h^{(1)}_{\mathbf{n}}$ becomes. Note, that in continuous Floquet
theory, all Fourier coefficients $\widetilde{\mathcal{H}}^{(\mathbf{n})}$
contribute to the first-order effective Hamiltonian for every value of
$\omega_{\mathbf{n}}$. Resonant terms ($\omega_{\mathbf{n}_{0}}=0$) contribute
strongly ($\text{sinc}(0)=1$) while non-resonant terms
($\omega_{\mathbf{n}}\neq 0$) contribute weakly with their weight given by the
value of the sinc function at frequency zero. In contrast, the first-order
Hamiltonian in traditional Floquet theory only contains resonant terms
$\widetilde{\mathcal{H}}^{(\mathbf{n}_{0})}$ ($\omega_{\mathbf{n}_{0}}=0$)
and, therefore, does not describe near-resonant behavior as we will see later
in various examples.

By taking the limit $T\rightarrow\infty$ of Eq. (18) and Eq. (19), we obtain:

$\lim\limits_{T\to\infty}h^{(1)}_{\mathbf{n}}(T)=\begin{dcases}1&\text{if}\quad\omega_{\mathbf{n}}=\omega_{\mathbf{n_{0}}}=0\\\
0&\text{else}\end{dcases}~{},$ (20)

which leads to the first-order effective Hamiltonian of single- and multi-mode
traditional Floquet theory:

$\lim\limits_{T\to\infty}\bar{{\mathcal{H}}}^{(1)}=\sum_{\mathbf{n_{0}}}\widetilde{\mathcal{H}}^{(\mathbf{n_{0}})}~{},$
(21)

For $T\rightarrow\infty$ the function $h^{(1)}_{\mathbf{n}}(T)$ is only non-
zero at the resonance conditions $\omega_{\mathbf{n_{0}}}=0$. Therefore the
sum of in Eq. (21) is only over multi-index $\mathbf{n_{0}}$, which fulfills
the resonance condition $\omega_{\mathbf{n_{0}}}=0$. As mentioned before, the
first-order effective Hamiltonians of traditional Floquet theory, only leads
to correct predictions at the resonance conditions
$\omega_{\mathbf{n_{0}}}=0$, but not for $\omega_{\mathbf{n_{0}}}\neq 0$. In
the following, we show that this is due to the requirement of periodic
Hamiltonians in traditional Floquet theory.

Figure 3: Plots of the function $h^{(1)}_{\mathbf{n}}(T)$. (a)
$h^{(1)}_{\mathbf{n}}(T)$ as function of the resonance $\omega_{\mathbf{n}}$
for different $T$. For $T\rightarrow\infty$ we obtain the result known from
traditional Floquet theory, depicted as a green star. (b)
$h^{(1)}_{\mathbf{n}}(T)$ as function of $\omega_{\mathbf{n}}$ and $T$. The
longer the duration $T$ becomes, the narrower the function
$h^{(1)}_{\mathbf{n}}(T)$, i.e., the more localized is the contribution of the
first-order Hamiltonian.

####  II.2.2 Second-order effective Hamiltonian

Let us now turn to the derivation of the closed-form expression of the second-
order Hamiltonian. Again, we use Eq. (17) to evaluate the integral in the
definition of the second-order effective Hamiltonian Eq. (2). For the second-
order effective Hamiltonian follows

$\displaystyle\bar{{\mathcal{H}}}^{(2)}$
$\displaystyle=-\frac{1}{2}\sum_{\mathbf{n},\mathbf{m}}[\widetilde{{\mathcal{H}}}^{(\mathbf{n})},\widetilde{{\mathcal{H}}}^{(\mathbf{m})}]\,h^{(2)}_{\mathbf{n},\mathbf{m}}(T)~{},$
(22)

with

$h^{(2)}_{\mathbf{n},\mathbf{m}}(T)=\frac{2}{T}\frac{\omega_{\mathbf{m}}\sin\left(\frac{\omega_{\mathbf{n}}T}{2}\right)\cos\left(\frac{\omega_{\mathbf{m}}T}{2}\right)-\omega_{\mathbf{n}}\cos\left(\frac{\omega_{\mathbf{n}}T}{2}\right)\sin\left(\frac{\omega_{\mathbf{m}}T}{2}\right)}{\omega_{\mathbf{n}}\omega_{\mathbf{m}}(\omega_{\mathbf{n}}+\omega_{\mathbf{m}})}~{}.$
(23)

The derivation of these equations can be found in the SI. Note, that the
singularities of the functions $h^{(2)}_{\mathbf{n},\mathbf{m}}(T)$ in
$\omega_{\mathbf{n}}$ and $\omega_{\mathbf{m}}$ are removable, i.e.,

$\displaystyle\lim_{\omega_{\mathbf{n}/\mathbf{m}}\to
0}h^{(2)}_{\mathbf{n},\mathbf{m}}(T)=\pm\frac{\omega_{\mathbf{n}/\mathbf{m}}T\cos\left(\frac{\omega_{\mathbf{n}/\mathbf{m}}T}{2}\right)-2\sin\left(\frac{\omega_{\mathbf{n}/\mathbf{m}}T}{2}\right)}{\omega_{\mathbf{n}/\mathbf{m}}^{2}}~{},$
(24)
$\displaystyle\lim_{\omega_{\mathbf{n}/\mathbf{m}}\to-\omega_{\mathbf{m}/\mathbf{n}}}h^{(2)}_{\mathbf{n},\mathbf{m}}(T)=\pm\frac{\sin(\omega_{\mathbf{m}/\mathbf{n}}T)-\omega_{\mathbf{m}/\mathbf{n}}T}{\omega_{\mathbf{m}/\mathbf{n}}^{2}}~{}.$
(25)

Figure 4 (a) shows $h^{(2)}_{\mathbf{n},-\mathbf{n}}(T)$ as a function of
$\omega_{\mathbf{n}}=-\omega_{\mathbf{m}}$ for different durations $T$. For
increasing duration the $h^{(2)}_{\mathbf{n},\mathbf{m}}(T)$ approximates
$\omega^{-1}_{\mathbf{n}}$. In contrast, for $T\rightarrow 0$,
$h^{(2)}_{\mathbf{n},\mathbf{m}}(T)$ approaches $0$ as required for a second-
order term. Similar to the first-order effective Hamiltonian, the second-order
effective Hamiltonian leads to a description of both resonant
($\omega_{\mathbf{n_{0}}}=0$) and non-resonant ($\omega_{\mathbf{n}}\neq 0$)
behavior. Again we can show the correspondence to traditional Floquet theory
by taking the limit $T\rightarrow\infty$:

$\lim\limits_{T\to\infty}h^{(2)}_{\mathbf{n},\mathbf{m}}(T)=\begin{dcases}{\omega_{\mathbf{n}}^{-1}}&\text{if}\quad\omega_{\mathbf{n}}=-\omega_{\mathbf{m}}\\\
0&\text{else}\end{dcases}~{},$ (26)

Hence we obtain

$\displaystyle\bar{{\mathcal{H}}}^{(2)}$
$\displaystyle=-\frac{1}{2}\sum_{\mathbf{n_{0}}}\sum_{\mathbf{n}\neq\mathbf{n_{0}}}\frac{[{\mathcal{H}}^{(\mathbf{n})},{\mathcal{H}}^{(\mathbf{n_{0}-n})}]}{\omega_{\mathbf{n}}}$
(27)

This is exactly the second-order effective Hamiltonian as obtained by
traditional operator-based Floquet theory, written with multi-index notation.
Note, that in the limit process the two multi-indices $\mathbf{n},\mathbf{m}$
present in Eq. (22) are reduced to only one, i.e., $\mathbf{n}$, for
traditional Floquet theory and that the resonance condition
$\omega_{\mathbf{n_{0}}}=0$ is excluded in the summation. However, in
continuous Floquet theory, $\omega_{\mathbf{n_{0}}}=0$ is not excluded but has
zero contribution due to the zero value of
$h^{(2)}_{\mathbf{n},\mathbf{-n}}(T)$.

Figure 4: Plots of the function $h^{(2)}_{\mathbf{n},\mathbf{m}}(T)$. (a)
$h^{(2)}_{\mathbf{n},-\mathbf{n}}(T)$ as function of the resonance
$\omega_{\mathbf{n}}=\omega_{-\mathbf{m}}$ for different $T$. Note, that for
$T\rightarrow\infty$ we obtain the result known from traditional Floquet
theory (green line). (b) $h^{(2)}_{\mathbf{n},\mathbf{m}}(T)$ as function of
$\omega_{\mathbf{n}}$ and $\omega_{\mathbf{m}}$ for $T=0.5\,$ms. Note, that
the function is anti-symmetric in respect to the
$\omega_{\mathbf{n}}=\omega_{\mathbf{m}}$ axis.

As mentioned a particular effective Hamiltonian corresponds to a specific
duration $T$. To change the duration of an effective Hamiltonian, we just vary
the parameter $T$, which only affects the functions $h_{\mathbf{n}}^{(1)}$ and
$h^{(2)}_{\mathbf{n,m}}$ of the first- and second-order effective Hamiltonian.
Hence the effective Hamiltonians of continuous Floquet theory are not linear
in $T$.

##  III Application

The following examples serve two purposes: firstly, as a practical guide on
how to apply continuous Floquet theory to various experimental schemes, and
secondly, to demonstrate the importance of the finite duration of pulse
sequences for the description of near-resonance behavior. The procedure for
applying continuous Floquet theory consists of four main steps:

  1. (i)

Define the extent of your spin system that is required for the experiment to
be described. One strategy is to start with a minimal configuration and
gradually adding complexity to study its effects. This approach offers
flexibility and reduces computational burden. Alternatively, one can begin
with a more general model and simplify iteratively. While more effort may be
required initially, it ensures a thorough understanding and adaptability to
different experimental conditions.

  2. (ii)

Identify and apply a suitable interaction-frame transformation Cho (1999);
Boutis  _et al._ (2003); Chávez and Ernst (2022b). This step allows us to
adapt the theory to the specific problem at hand by selecting an interaction
frame that is most suitable for observing the dynamics of interest and
simplifying the Hamiltonian. In many cases, this is the rf irradiation and can
in addition include the chemical shift.

  3. (iii)

Choose a set of characteristic frequencies to formulate a periodic
continuation of the Hamiltonian as a Fourier series. In this step, there is
flexibility to customize the description according to the system of interest
by selecting appropriate characteristic frequencies for the Fourier series
expansion of the Hamiltonian. As a default option, continuation with the
frequency $\omega_{\mathrm{p}}=2\pi/T$ (Eq. (9)) can always be used.

  4. (iv)

Insert the Fourier coefficients of the Hamiltonian into the expressions of the
effective Hamiltonians. This step involves substituting the Fourier
coefficients of the Hamiltonian into the definition of the effective
Hamiltonians (Eqs. (18) and (22)).

The general Hamiltonian in solid-state NMR consists of a
$\textrm{I}_{N_{I}}\textrm{S}_{N_{S}}$ system of spin-1/2 which can be
described by a time-dependent Hamiltonian of the form:

$\displaystyle{\mathcal{H}}(t)$
$\displaystyle={\mathcal{H}}_{\textrm{I}\textrm{I}}(t)+{\mathcal{H}}_{\textrm{I}\textrm{S}}(t)+{\mathcal{H}}_{\textrm{S}\textrm{S}}(t)$
(28)
$\displaystyle\quad+{\mathcal{H}}_{\textrm{I}}(t)+{\mathcal{H}}_{\textrm{S}}(t)+{\mathcal{H}}_{\mathrm{rf}}(t)~{}.$
(29)

The chemical shifts are characterized with

$\displaystyle{\mathcal{H}}_{\textrm{I}}(t)$
$\displaystyle=\sum\limits_{p}\sum\limits_{n=-2}^{2}\omega_{\textrm{I}_{p}}^{(n)}e^{in\omega_{\textrm{r}}t}\textrm{I}_{pz}~{},$
(30) $\displaystyle{\mathcal{H}}_{\textrm{S}}(t)$
$\displaystyle=\sum\limits_{p}\sum\limits_{n=-2}^{2}\omega_{\textrm{S}_{p}}^{(n)}e^{in\omega_{\textrm{r}}t}\textrm{S}_{pz}~{},$
(31)

where $n=0$ describes to the isotropic and $n\neq 0$ the anisotropic chemical
shift. The homonuclear couplings between the I and S spins are given by

$\displaystyle{\mathcal{H}}_{\textrm{I}\textrm{I}}(t)$
$\displaystyle=\sum_{p<q}\biggl{[}\omega_{\textrm{I}_{p}\textrm{I}_{q}}^{(0)}\,2\,\vec{\textrm{I}}_{p}\cdot\vec{\textrm{I}}_{q}$
$\displaystyle\quad\quad\quad+\sum\limits_{\begin{subarray}{c}n=-2\\\ n\neq
0\end{subarray}}^{2}\omega_{\textrm{I}_{p}\textrm{I}_{q}}^{(n)}e^{in\omega_{\textrm{r}}t}\,(3\,\textrm{I}_{pz}\textrm{I}_{qz}-\vec{\textrm{I}}_{p}\cdot\vec{\textrm{I}}_{q})\biggr{]}~{},$
(32) $\displaystyle{\mathcal{H}}_{\textrm{S}\textrm{S}}(t)$
$\displaystyle=\sum_{u<v}\biggl{[}\omega_{\textrm{S}_{u}\textrm{S}_{v}}^{(0)}\,2\,\vec{\textrm{S}}_{u}\cdot\vec{\textrm{S}}_{v}$
$\displaystyle\quad\quad\quad+\sum\limits_{\begin{subarray}{c}n=-2\\\ n\neq
0\end{subarray}}^{2}\omega_{\textrm{S}_{u}\textrm{S}_{v}}^{(n)}e^{in\omega_{\textrm{r}}t}\,(3\,\textrm{S}_{uz}\textrm{S}_{vz}-\vec{\textrm{S}}_{u}\cdot\vec{\textrm{S}}_{v})\biggr{]}~{}.$
(33)

The $\omega_{\textrm{I}_{p}\textrm{I}_{q}}^{(0)}$ corresponds to the isotropic
$J$ coupling and $\omega_{\textrm{I}_{p}\textrm{I}_{q}}^{(n)}$ to the
anisotropic dipolar coupling.

$\displaystyle{\mathcal{H}}_{\textrm{I}\textrm{S}}(t)$
$\displaystyle=\sum\limits_{p,u}\sum\limits_{n=-2}^{2}\omega_{\textrm{I}_{p}\textrm{S}_{u}}^{(n)}e^{in\omega_{\textrm{r}}t}\,2\,\textrm{I}_{pz}\textrm{S}_{uz}$
(34)

characterizes the isotropic ($n=0$) and an isotropic ($n\neq 0$) heteronuclear
coupling. Expressions for the various $\omega^{(n)}$ parameters can be found
in Ref. Scholz, Beek, and Ernst (2010) Eqs. (34) and (35). Finally,

$\displaystyle{\mathcal{H}}_{\mathrm{rf}}(t)$
$\displaystyle=\omega_{1\textrm{I}}(t)\sum\limits_{p}(\textrm{I}_{px}\cos(\phi_{\textrm{I}}(t))+\textrm{I}_{py}\sin(\phi_{\textrm{I}}(t)))$
$\displaystyle\quad+\omega_{1\textrm{S}}(t)\sum\limits_{u}(\textrm{S}_{ux}\cos(\phi_{\textrm{S}}(t))+\textrm{S}_{uy}\sin(\phi_{\textrm{S}}(t)))$
(35)

describes arbitrary rf irradiation on the I and S spins, respectively.

###  III.1 Recoupling Using Constant RF Irradiation Under MAS

NMR experiments often involve applying a constant rf irradiation during a time
$T$ to a single spin species, such as in recoupling experiments like
homonuclear rotary resonance (HORROR) Nielsen  _et al._ (1994) or rotary-
resonance recoupling ($\mathrm{R}^{3}$) Levitt, Oas, and Griffin (1988); Oas,
Griffin, and Levitt (1988), as well as non-resonant continuous wave (CW)
decoupling experiments Ernst, Samoson, and Meier (2005). In the following, we
will consider the Hamiltonian of Eq. (28) with a constant rf irradiation on
the I spin with duration $T$, i.e.,

${\mathcal{H}}_{\textrm{rf}}(t)=\omega_{1}\textrm{I}_{x}\,\Pi(t/T)~{}.$ (36)

As a first step, we simplify the Hamiltonian of Eq. (28) by going into an
interaction frame with the rf irradiation mediated by the unitary operator

$\textrm{U}(t)=\exp\left(i\omega_{1}t\sum_{p}\textrm{I}_{pz}\right)\exp\left(-i\frac{\pi}{2}\sum_{p}\textrm{I}_{py}\right)~{}.$
(37)

This operator first rotates the Hamiltonian by 90∘ about the $-y$ axis
$(\textrm{I}_{px}\rightarrow\textrm{I}_{pz},\textrm{I}_{pz}\rightarrow-\textrm{I}_{px})$
and subsequently transforms the Hamiltonian with the radio-frequency part of
the Hamiltonian into an interaction frame. Note, that alternatively the
chemical-shift term can also be included in the interaction-frame
transformation. In this case, the chemical-shift term is eliminated but
manifests as a change in the modulation of the coupling terms. Hence, on one
hand, the Hamiltonian becomes simpler because only the interaction-frame
modulated coupling terms are left, but on the other hand, it adds complexity
to the time dependence of the coupling terms. Here, the chemical shift is not
included to prevent obfuscation of the simple modulation of the coupling terms
caused by the rf irradiation. Note that for complex rf irradiations or to
study the chemical-shift dependence, it is often advisable to eliminate the
chemical-shift term by including it in the interaction frame. After the
interaction-frame transformation, we can identify two intrinsic frequencies,
namely $\omega_{\textrm{r}}$ and $\omega_{1}$, which naturally serve as
suitable options for the Fourier series expansion of the Hamiltonian. In
principle, all terms in the interaction frame should be marked by a ′ to
distinguish them from the standard rotating frame but for ease of notation, we
drop the ′ again after Eq. (III.1).

$\displaystyle{\mathcal{H}}^{\prime}(t)$
$\displaystyle=\textrm{U}(t){\mathcal{H}}(t)\textrm{U}^{\dagger}(t)$
$\displaystyle=\sum_{n=-2}^{2}\sum_{k=-2}^{2}{\mathcal{H}}^{\prime(n,k)}\,e^{in\omega_{\mathrm{r}}t}\,e^{ik\omega_{1}t}\,\Pi(t/T)~{},$
(38)

with the Fourier coefficients

$\displaystyle{{\mathcal{H}}}^{(0,0)}$
$\displaystyle=\sum_{u}\omega_{\textrm{S}_{u}}^{(0)}\textrm{S}_{uz}+\sum_{p<q}\omega_{\textrm{I}_{p}\textrm{I}_{q}}^{(0)}2\vec{\textrm{I}}_{p}\cdot\vec{\textrm{I}}_{q}$
$\displaystyle\quad+\sum_{u<v}\omega_{\textrm{S}_{u}\textrm{S}_{v}}^{(0)}2\vec{\textrm{S}}_{u}\cdot\vec{\textrm{S}}_{v}~{},$
(39) $\displaystyle{{\mathcal{H}}}^{(n,0)}$
$\displaystyle=\sum_{u}\omega_{S_{u}}^{(n)}\textrm{S}_{uz}-\frac{1}{2}\sum_{p<q}\omega_{\textrm{I}_{p}\textrm{I}_{q}}^{(n)}\left[3{\textrm{I}}_{pz}{\textrm{I}}_{qz}-\vec{\textrm{I}}_{p}\cdot\vec{\textrm{I}}_{q}\right]$
$\displaystyle\quad+\sum_{u<v}\omega_{\textrm{S}_{u}\textrm{S}_{v}}^{(n)}\biggl{[}3{\textrm{S}}_{uz}{\textrm{S}}_{vz}-\vec{\textrm{S}}_{u}\cdot\vec{\textrm{S}}_{v}\biggr{]}~{},$
(40) $\displaystyle{{\mathcal{H}}}^{(n,\pm 1)}$
$\displaystyle=-\sum_{p}\frac{\omega_{\textrm{I}_{p}}^{(n)}}{2}{\textrm{I}}_{p}^{\pm}-\sum_{pu}\omega^{(n)}_{\textrm{I}_{p}\textrm{S}_{u}}\textrm{I}_{pz}\textrm{S}^{\pm}_{u}~{},$
(41) $\displaystyle{{\mathcal{H}}}^{(n,\pm 2)}$
$\displaystyle=-\sum_{p<q}\frac{3}{4}\omega_{\textrm{I}_{p}\textrm{I}_{q}}^{(n)}{\textrm{I}}^{\pm}_{p}{\textrm{I}}^{\pm}_{q}~{}.$
(42)

From Eq. (18) we directly obtain the first-order Hamiltonian

$\bar{{\mathcal{H}}}^{(1)}=\sum_{n,k}{\mathcal{H}}^{(n,k)}h^{(1)}_{(n,k)}(T)~{},$
(43)

with

$h^{(1)}_{(n,k)}(T)=\text{sinc}\left(\frac{(n\omega_{\textrm{r}}+k\omega_{1})T}{2}\right)~{}.$
(44)

The second-order effective Hamiltonian follows directly from Eq. (22)

$\displaystyle\bar{{\mathcal{H}}}^{(2)}=-\frac{1}{2}\sum_{\begin{subarray}{c}n_{1},k_{1}\\\
n_{2},k_{2}\end{subarray}}[{\mathcal{H}}^{(n_{1},k_{1})},{\mathcal{H}}^{(n_{2},k_{2})}]h^{(2)}_{(n_{1},k_{1}),(n_{2},k_{2})}(T)~{},$
(45)

with

$h^{(2)}_{(n_{1},k_{1}),(n_{2},k_{2})}(T)=\frac{2}{T}\frac{\omega_{n_{2},k_{2}}\sin\left(\frac{\omega_{n_{1},k_{1}}T}{2}\right)\cos\left(\frac{\omega_{n_{2},k_{2}}T}{2}\right)-\omega_{n_{1},k_{1}}\cos\left(\frac{\omega_{n_{1},k_{1}}T}{2}\right)\sin\left(\frac{\omega_{n_{2},k_{2}}T}{2}\right)}{\omega_{n_{1},k_{1}}\omega_{n_{2},k_{2}}(\omega_{n_{1},k_{1}}+\omega_{n_{2},k_{2}})}$
(46)

where $\omega_{n_{1},k_{1}}=(n_{1}\omega_{\textrm{r}}+k_{1}\omega_{1})$ and
$\omega_{n_{2},k_{2}}=(n_{2}\omega_{\textrm{r}}+k_{2}\omega_{1})$. In
classical Floquet theory, only the resonant terms fulfilling the conditions
$n_{0}\omega_{\mathrm{r}}+k_{0}\omega_{1}=0$ are included in the first-order
effective Hamiltonian Ernst, Samoson, and Meier (2005), since all other terms
do not contribute. For continuous Floquet theory, all Fourier coefficients
will contribute to the first-order effective Hamiltonian with the magnitude
determined by the width of the sinc function and the distance of the frequency
$n\omega_{\mathrm{r}}+k\omega_{1}$ from the frequency 0. Therefore, for fast
MAS, the terms $\widetilde{{{\mathcal{H}}}}^{(n,0)}$ never contribute
significantly to the first-order effective Hamiltonian while the term
$\widetilde{{{\mathcal{H}}}}^{(0,0)}$ always contributes but is typically
small and can often be neglected. In the following, we will consider the
recoupling effects governed by the resonance conditions:

  1. (i)

At $\omega_{1}=\omega_{\textrm{r}}/2$ only the homonuclear dipolar coupling is
recoupled. This condition is called the HORROR condition Nielsen  _et al._
(1994), and the most significant Fourier coefficients here are
${\mathcal{H}}^{(0,0)},{\mathcal{H}}^{(1,-2)}$ and ${\mathcal{H}}^{(-1,2)}$.

  2. (ii)

At $\omega_{1}=\omega_{\textrm{r}}$ the CSA tensor, the heteronuclear dipolar
and the homonuclear dipolar coupling are recoupled Levitt, Oas, and Griffin
(1988); Oas, Griffin, and Levitt (1988). Hence the relevant Fourier
coefficients are ${\mathcal{H}}^{(0,0)}$, ${\mathcal{H}}^{(1,-1)}$,
${\mathcal{H}}^{(-1,1)}$, ${\mathcal{H}}^{(2,-2)}$ and
${\mathcal{H}}^{(-2,2)}$.

  3. (iii)

At $\omega_{1}=2\omega_{\textrm{r}}$ the CSA tensor and the heteronuclear
dipolar coupling Levitt, Oas, and Griffin (1988); Oas, Griffin, and Levitt
(1988) are recoupled. Here the Fourier coefficients ${\mathcal{H}}^{(0,0)}$,
${\mathcal{H}}^{(2,-1)}$ and ${\mathcal{H}}^{(-2,1)}$ dominate.

Condition (ii) and (iii) are rotary-resonance conditions, which include all
conditions of the form $\omega_{1}=n\omega_{\textrm{r}}$ with
$n\in\mathbb{N}$. The higher-order rotary-resonance conditions ($n\geq 3$),
however, are less relevant because they are much weaker and difficult to
observe experimentally. Furthermore, also fractional rotary resonance
condition exist at $\omega_{1}=k\omega_{\textrm{r}}/n$ which are usually also
relatively week Beek  _et al._ (2011). Because the Fourier coefficient
${\mathcal{H}}^{(0,0)}$ is typically weak, ${\mathcal{H}}^{(n,\pm 1)}$ and
${\mathcal{H}}^{(n,\pm 2)}$ characterize the time evolution caused by the
irradiation. The first one represents the recoupling of the CSA and
heteronuclear dipolar coupling, while the second one includes the DQ
recoupling of the homonuclear dipolar coupling.

####  III.1.1 HORROR

Fig. 5 shows the comparison between numerical time-slicing (exact) simulation
and effective Hamiltonian simulation of the transfer intensity
$\textrm{I}_{1x}(0)\longrightarrow\textrm{I}_{2x}(T)$ (in the laboratory
frame) for a homonuclear two-spin system as a function of resonance mismatch
$\nu_{1}/\nu_{\textrm{r}}$ for the durations $T=0.3\,$ms, $T=0.5\,$ms and
$T=1\,$ms. The MAS frequency $\nu_{\textrm{r}}=100\,$kHz has been kept
constant, whereas the rf-field amplitude $\nu_{1}$ was varied. The initial
density matrix was set to $\rho_{0}=\textrm{I}_{1x}$ (along the rf field) and
detecting $\textrm{I}_{2x}$ after the duration $T$. Different crystallites
were simulated with 300 spherical-powder grid orientations for powder
averaging Suzukawa  _et al._ (1973). The dipolar coupling strength was set to
$\delta_{\mathrm{CC}}/(2\pi)=-4.5\,$kHz corresponding to a typical one-bond CC
coupling. Due to the double-quantum nature of the polarization transfer, the
transferred polarization appears with negative intensity. For $T=0.3\,$ms the
exact transfer intensity $\langle\textrm{I}_{2x}\rangle$ is reproduced by the
first-order effective Hamiltonian (Fig. 5a), with no visible contribution from
the second-order effective Hamiltonian (Fig. 5d). $T=0.4\,$ms slightly exceeds
the time needed to achieve maximal recoupling, leading to minor deviations
between the recoupling intensity derived by exact and first-order effective
Hamiltonian (Fig. 5b). However, the simulation up to the second-order
effective Hamiltonian (Fig. 5e) accurately reproduces the transfer intensity.
Note, that the time needed to achieve maximal recoupling is approximately
proportional to the inverse of the dipolar coupling strength. $T=1\,$ms
surpasses the time needed to achieve maximum recoupling intensity, leading to
deviations between exact and first-order effective Hamiltonian simulation
close to the resonance condition (Fig. 5c). Using the first and second-order
effective Hamiltonian leads to more accurate simulation (Fig. 5f) of the
transfer intensity for $T=1\,$ms than simulations using only the first-order
Hamiltonian. The recoupling bandwidth is in good approximation proportional to
$T^{-1}$, hence decreasing the duration $T$ leads to a larger bandwidth of the
transfer intensity. Note, that on the resonance condition, the first-order
effective Hamiltonian always reproduces the transfer intensity for all mixing
times.

Figure 5: Comparison between numerical time-slicing (exact) and effective
Hamiltonian simulation of the polarization-transfer intensity
$\langle\textrm{I}_{2x}\rangle$ near the HORROR resonance condition
$\nu_{1}/\nu_{r}=0.5$ for different durations $T$. Different crystallites were
simulated with 300 spherical-powder grid orientations for powder averaging
Suzukawa  _et al._ (1973). For a standard homonuclear one-bond CC coupling,
the dipolar coupling strength was set to
$\delta_{\mathrm{CC}}/(2\pi)=-4.5\,$kHz. Subfigures (a)-(c) show the
comparison between the recoupling intensity $\langle\textrm{I}_{2x}\rangle$
obtained from the exact and first-order effective Hamiltonian simulation. For
durations, longer than the duration needed to achieve maximal transfer
intensity ((b) and (c)), the agreement with the exact calculation decreases
with increasing duration, especially near the resonance condition. Subfigures
(d)-(f) show the comparison between the recoupling intensity
$\langle\textrm{I}_{2x}\rangle$ obtained from exact and up to second-order
effective Hamiltonian simulation. For times up $0.5\,$ms the second-order
effective Hamiltonian has almost zero contribution to the transfer intensity
$\langle\textrm{I}_{2x}\rangle$. For large durations, greatly exceeding the
time needed to achieve the maximal recoupling intensity ((c) and (f)),
including the second-order effective Hamiltonian significantly the agreement
with exact simulations compared to calculations using only the first-order
effective Hamiltonian.

####  III.1.2 Rotary-Resonance Recoupling

In Fig. 6 exact and effective Hamiltonian simulations are compared for the
rotary resonance conditions $\omega_{1}=n\omega_{\textrm{r}}$ with
$n\in\\{1,2\\}$ for a heteronuclear I-S two-spin system with irradiation on
the I-spin. In such an experiment, a dephasing of the magnetization on the
S-spin is observed due to the reintroduction of the heteronuclear dipolar
coupling. As in Fig. 5 the MAS frequency $\nu_{\textrm{r}}$ was kept constant
at 100 kHz while the rf-field amplitude $\nu_{1}$ was varied to map out the
resonance conditions. The simulation parameters have been kept consistent with
the previous simulation for the HORROR recoupling, except the dipolar coupling
strength was set to $\delta_{\mathrm{HC}}/(2\pi)=-46\,$kHz corresponding to a
typical heteronuclear one-bond CH coupling. Similar to HORROR recoupling the
first-order effective Hamiltonian reproduces the dephasing accurately for
durations until the maximal transfer intensity is reached (Figs. 6 (a)-(c).
For durations exceeding the first maximum of the polarization transfer,
higher-order corrections are necessary and become more important for
increasing dephasing durations as can be seen in Figs. 6 (d)-(f).

Figure 6: Comparison between exact and effective Hamiltonian simulations of
the dephasing of $\textrm{S}_{x}$ of a heteronuclear two-spin system as a
function of $\nu_{1}/\nu_{r}$ for different durations $T$. As before, 300
spherical-powder grid orientations were used for powder averaging Suzukawa
_et al._ (1973). The dipolar coupling strength was set to
$\delta_{\mathrm{CH}}/(2\pi)=-46\,$kHz, corresponding to a standard
heteronuclear one-bond CH coupling. Subfigures (a)-(c) illustrate the
comparison between exact and first-order effective Hamiltonian simulations,
while (d)-(f) include second-order corrections. The recoupling bandwidth for
the rotary resonance conditions $\omega_{1}=n\omega_{\textrm{r}}$ is
approximately proportional to $T^{-1}$. For large durations, greatly exceeding
the time needed to achieve the maximal dephasing ((c) and (f)), second-order
corrections become significant.

###  III.2 C-type and R-type Pulse Schemes

Symmetry-based $\mathrm{C}N_{n}^{\nu}$ and $\mathrm{R}N_{n}^{\nu}$ pulse
schemes constitute an important class of rotor-synchronized pulse schemes used
for recoupling in solid-state NMR. A substantial body of literature has been
dedicated to symmetry-based pulse schemes, including consideration based on
first and second-order average Hamiltonian Levitt (2002); Brinkmann and Edén
(2004); Vinogradov, Madhu, and Vega (2004b). $\mathrm{C}N_{n}^{\nu}$ and
$\mathrm{R}N_{n}^{\nu}$ pulse schemes consist of basic elements that are
synchronized with the MAS frequency. Depending on the symmetry coefficients
($N,n,\nu$) such pulse schemes lead to selective re- or decoupling of specific
interactions. The coefficient $N$ represents the number of basic elements
arranged within a specified number of rotor cycles denoted by $n$. The phase
of each basic element is $\phi=2\pi\nu/N$ in the case of
$\mathrm{C}N_{n}^{\nu}$ and $\phi=\pi\nu/N$ in the case of
$\mathrm{R}N_{n}^{\nu}$ pulse schemes. The rf Hamiltonian for such phase-
modulated pulse schemes can be written as

${\mathcal{H}}_{\textrm{rf}}(t)=\omega_{1}\sum_{u}\biggl{(}\textrm{S}_{ux}\cos(\phi(t))+\textrm{S}_{uy}\sin(\phi(t))\biggr{)}~{}.$
(47)

If we only include the rf Hamiltonian in the interaction-frame transformation
the spin part of the resulting Hamiltonian is periodic with the modulation
frequency of the pulse scheme $\tau_{\mathrm{m}}$, which is the duration of
the complete symmetry-based sequence. Because the spatial part of the
Hamiltonian is periodic with the MAS frequency $\omega_{\textrm{r}}$, we can
express the resulting Hamiltonian with Fourier series in the frequencies
$\omega_{\mathrm{r}}$ and $\omega_{\mathrm{m}}$:

${\mathcal{H}}(t)=\sum_{n,k}\widetilde{{\mathcal{H}}}^{(n,k)}e^{in\omega_{\mathrm{r}}t}e^{ik\omega_{\mathrm{m}}t}\Pi(t/T)~{}.$
(48)

The Fourier coefficients of the Hamiltonian are Vinogradov, Madhu, and Vega
(2004b); Scholz, Beek, and Ernst (2010):

$\displaystyle\widetilde{{\mathcal{H}}}^{(0,k)}$
$\displaystyle=\left\\{\sum_{p<q}\omega_{\mathrm{I}_{p}\mathrm{I}_{q}}^{(0)}2\vec{\textrm{I}}_{p}\cdot\vec{\textrm{I}}_{q}+\sum_{u<v}\omega_{\mathrm{S}_{u}\mathrm{S}_{v}}^{(0)}2\vec{\textrm{S}}_{u}\cdot\vec{\textrm{S}}_{v}+\sum_{p}\omega_{\mathrm{I}_{p}}^{(0)}\textrm{I}_{pz}\right\\}\delta_{k,0}+\sum_{u}\omega_{\mathrm{S}_{u}}^{(0)}\sum_{s=-1}^{1}a_{1,s}^{(k)}T_{1,s}^{(u)}$
$\displaystyle\quad+\sum_{p,u}\omega_{\textrm{I}_{p}\textrm{S}_{u}}^{(0)}2\textrm{I}_{pz}\sum_{s=-1}^{1}a_{1,s}^{(k)}T_{1,s}^{(u)}~{},$
(49) $\displaystyle\widetilde{{\mathcal{H}}}^{(n,k)}$
$\displaystyle=\left\\{\sum_{p<q}\omega_{\textrm{I}_{p}\textrm{I}_{q}}^{(n)}\left(3\textrm{I}_{pz}\textrm{I}_{qz}-\vec{\textrm{I}}_{p}\cdot\vec{\textrm{I}}_{q}\right)+\sum_{p}\omega_{\textrm{I}_{p}}^{(n)}\textrm{I}_{pz}\right\\}\delta_{k,0}+\sum_{u<v}\sqrt{6}\,\omega_{\textrm{S}_{u}\textrm{S}_{v}}^{(n)}\sum_{s=-2}^{2}a_{2,s}^{(k)}T_{2,s}^{(u,v)}$
$\displaystyle\quad+\sum_{u}\omega_{\textrm{S}_{u}}^{(n)}\sum_{s=-1}^{1}a_{1,s}^{(k)}T_{1,s}^{(u)}+\sum_{p,d}\omega_{\textrm{I}_{p}\textrm{S}_{u}}^{(n)}2\textrm{I}_{pz}\sum_{s=-1}^{1}a_{1,s}^{(k)}T_{1,s}^{(u)}~{},$
(50)

where

$a_{r,s}^{(k)}=\frac{1}{T}\int_{-T/2}^{T/2}a_{r,s}(t)e^{-ik\omega_{\mathrm{m}}t}\textrm{d}t~{}.$
(51)

are the Fourier coefficients of the interaction-frame transformation of the
$T_{r,s}^{(p)}$ operator. The expression for $T_{r,s}^{(p)}$ can be located in
Chapter 2, Table 2.3 of the book by Mehring Mehring (1983). To obtain the
effective Hamiltonian we simply insert Eq. (48) into Eq. (18) and Eq. (22). An
alternative approach is to include the chemical-shift Hamiltonian in the
interaction-frame transformation together with the rf Hamiltonian. One notable
advantage of this approach is that the chemical-shift terms in the Hamiltonian
Eq. (48) are eliminated because the chemical-shift and the rf Hamiltonian are
encoded in the interaction frame trajectory $a_{r,s}(t)$. However, in this
interaction frame the effective field is not zero and therefore the
interaction-frame trajectory $a_{r,s}(t)$ lacks periodicity with
$\omega_{\mathrm{m}}$. Hence utilizing a Fourier series solely with
$\omega_{\mathrm{m}}$ is not possible, but $a_{r,s}(t)$ can be expressed using
the frequencies $\omega_{\mathrm{m}}$ and $\omega_{\textrm{eff}}$, i.e.

$a_{r,s}^{(k,\ell)}=\frac{1}{T}\int_{-T/2}^{T/2}a_{r,s}(t)e^{-ik\omega_{\mathrm{m}}t}e^{-i\ell\omega_{\mathrm{eff}}t}\textrm{d}t~{}.$
(52)

The Hamiltonian is therefore written in terms of three characteristic
frequencies $\omega_{\textrm{r}}$, $\omega_{\mathrm{m}}$ and
$\omega_{\textrm{eff}}$:

$\widetilde{{\mathcal{H}}}(t)=\sum_{n,k,l}\widetilde{{\mathcal{H}}}^{(n,k,l)}e^{i(n\omega_{\textrm{r}}+k\omega_{\mathrm{m}}+\ell\omega_{\textrm{eff}})t}~{}.$
(53)

As a result, the resonance conditions are
$n\omega_{\textrm{r}}+k\omega_{\mathrm{m}}+\ell\omega_{\textrm{eff}}$ with
$n\in\\{-2,-1,..,2\\}$, $k\in\mathbb{R}$ and $\ell\in\\{-1,0,1\\}$. As before
the first and second-order effective Hamiltonian are calculated using Eq. (18)
and Eq. (22). The two approaches, Eq. (48) and Eq. (53), lead to similar
effective Hamiltonians, however including the chemical shift into the
interaction frame transformation as for Eq. (53) can lead to faster
convergence of the effective Hamiltonian than Eq. (48) as described in Chávez
and Ernst (2022b). Note, that the two approaches only differ if chemical
shifts are considered.

####  III.2.1 C-type Pulse Schemes

Probably one of the most popular and historic $\mathrm{C}$-type pulse schemes
is $\mathrm{C}7_{n}^{1}$, where seven basic building blocks, comprising
$(2\pi)_{\phi},(2\pi)_{\phi+\pi}$, are applied within $n$ rotor periods. Fig.
7 shows the efficiency of the polarization transfer
$\textrm{S}_{1z}\rightarrow\textrm{S}_{2z}$ by plotting
$\langle\textrm{S}_{2z}\rangle$ comparing exact numerical simulation and the
time evolution under a first- and second-order effective Hamiltonian for
$\mathrm{C}7_{n}^{1}$ with different mixing times $T$. A homonuclear two-spin
system was used with a dipolar coupling strength of
$\delta_{\mathrm{CC}}/(2\pi)=-4.5\,$kHz corresponding to a typical one-bond CC
coupling. As before, 300 spherical powder grid orientations were simulated for
powder averaging Suzukawa  _et al._ (1973). The rf amplitude $\nu_{1}=70\,$kHz
has been kept constant whereas the MAS frequency $\nu_{r}$ was varied.
The three areas of high polarization transfer in Fig. 7 correspond to the
symmetry sequences (from left to right) $\mathrm{C}7^{1}_{4}$,
$\mathrm{C}7^{1}_{2}$ and $\mathrm{C}28^{4}_{5}$. The peak associated with
$\mathrm{C}7^{1}_{2}$ achieves maximum recoupling intensity at the fastest
rate and exhibits the broadest width. Subsequently, the $\mathrm{C}7^{1}_{4}$
peak, being the second broadest, follows, while the narrowest peak corresponds
to $\mathrm{C}28^{4}_{5}$ reflecting different scaling factors for the dipolar
coupling. As in the previous examples, the exact recoupling intensity is
replicated using only the first-order effective Hamiltonian for durations $T$
up to the time required to obtain the maximum transfer intensity. For
durations exceeding the maximum transfer intensity, the second-order
correction becomes important. Corrections above second-order are only
important when the duration $T$ is much bigger than the time needed to achieve
maximum transfer.

Figure 7: Transferred magnetization $\langle\textrm{S}_{2z}\rangle$ under
$\mathrm{C}$-type pulse schemes obtained from exact calculations and effective
Hamiltonian calculations. The three visible peaks correspond to
$\mathrm{C}7^{1}_{4}$, $\mathrm{C}7^{1}_{2}$ and $\mathrm{C}28^{4}_{5}$,
reading form left to right. Subfigures (a)-(c) show the comparison between
exact and first-order effective Hamiltonian simulations, for different
durations $T$. The comparison between exact and up to second-order effective
Hamiltonian simulation is depicted in (d)-(f). The rf amplitude
$\nu_{1}=70\,$kHz has been kept constant whereas the MAS frequency $\nu_{r}$
was varied. As in the previous examples, different crystallites with 300
spherical powder grid orientations were simulated for powder averaging
Suzukawa  _et al._ (1973). The dipolar coupling strength was set to
$\delta_{\mathrm{CC}}/(2\pi)=-4.5\,$kHz.

An important property of recoupling pulse schemes is the sensitivity to the
chemical-shift offset of the irradiated spin $\nu_{\mathrm{S}}^{(0)}$.
Ideally, these schemes should exhibit effective recoupling across a broad
range of chemical-shift offsets. To extend this range the basic element of the
$\mathrm{C}N_{n}^{\nu}$ pulse schemes $(2\pi)_{\phi},(2\pi)_{\phi+\pi}$ is
substituted with the composite pulse
$(\pi/2)_{\phi},(2\pi)_{\phi+\pi},(3\pi/2)_{\phi}$ Hohwy  _et al._ (1998).
These modified $\mathrm{C}$ pulse schemes, termed
POST-$\mathrm{C}N_{n}^{\nu}$, are commonly used in experimental
implementations due to their better offset compensation. Figure 8 shows the
transferred polarization as a function of the chemical-shift offset
$\nu^{(0)}_{\textrm{S}}$ for the $\mathrm{C}7^{1}_{2}$ and
POST-$\mathrm{C}7^{1}_{2}$ pulse scheme. As expected, the
POST-$\mathrm{C}7^{1}_{2}$ pulse scheme exhibits a better chemical-shift
offset bandwidth of approx. $0.8\nu_{1}=56\,$kHz, compared to the standard
$\mathrm{C}7^{1}_{2}$ pulse scheme with a bandwidth of approx.
$0.3\nu_{1}=21\,$kHz. The simulation parameters, including the spin system
parameters and rf amplitude, remain consistent with those utilized in Fig. 7.
The effective Hamiltonians were calculated using Eq. (53), thus in the
interaction frame including the rf and the chemical-shift Hamiltonian.

Figure 8: Recoupling intensity $\langle\textrm{S}_{2z}\rangle$ for the
$\mathrm{C}7^{1}_{2}$ and POST-$\mathrm{C}7^{1}_{2}$ pulse scheme obtained
from exact calculations and effective Hamiltonian calculations up to second
order as a function of $\nu^{(0)}_{\textrm{S}}/\nu_{1}$. Subfigures (a)-(c)
correspond to $\mathrm{C}7^{1}_{2}$ leading to a recoupling bandwidth of
approximately $0.3\nu_{1}=21\,$kHz. In contrast, subfigures (d)-(f) show the
performance of the POST-$\mathrm{C}7^{1}_{2}$ pulse scheme, which exhibits a
wider recoupling bandwidth of approx. $0.8\nu_{1}=56\,$kHz. The simulation
parameters are consistent with Fig. 7.

####  III.2.2 R-type Pulse Schemes

Typical implementation of the basic element of $\mathrm{R}$-type pulse schemes
are either $(\pi)_{\phi},(\pi)_{-\phi}$, or using composite pulse
$(\pi/2)_{\phi}(3\pi/2)_{\phi+\pi}$ for the basic $\pi$ rotation.
Implementation using composite pulses increases the chemical-shift
compensation and, therefore, the recoupling bandwidth of the sequence while at
the same time requiring a higher rf-field amplitude at the same spinning
frequency. Fig. 9 shows the transfer intensity $\langle\textrm{S}_{2z}\rangle$
comparing exact and first and second-order effective Hamiltonian simulations
as a function of $\nu_{1}/\nu_{r}$ for $\mathrm{R}26_{8}^{11}$ (left peak) and
$\mathrm{R}26_{4}^{11}$ (right peak) for different $T$. The parameters for the
simulations have been kept consistent with those used for the previous
simulation of the C-type pulse schemes. As in the previous cases, the first-
order Hamiltonian is sufficient to reproduce the transfer intensity $\langle
S_{2z}\rangle$ for durations up to the point where maximal transfer intensity
is attained. For longer durations, second-order corrections become necessary.

Figure 9: Recoupling intensity $\langle\textrm{S}_{2z}\rangle$ under
$\mathrm{R}$-type pulse schemes obtained from exact calculations and effective
Hamiltonian calculations. The left peak corresponds to the
$\mathrm{R}26^{11}_{8}$ whereas the right peak corresponds to
$\mathrm{R}26^{11}_{4}$. Subfigures (a)-(c) show the comparison between exact
and first-order effective Hamiltonian simulations, for different durations
$T$. The comparison between exact and up to second-order effective Hamiltonian
simulation is represented in subfigure (d)-(f). The rf amplitude
$\nu_{1}=70\,$kHz has been kept constant whereas the MAS frequency $\nu_{r}$
was varied. The initial density matrix was set to $\textrm{I}_{1z}$ and after
the duration $T$ $\textrm{I}_{2z}$ was detected.

Fig. 10 compares the transfer intensity for the $\mathrm{R}26^{11}_{4}$ pulse
scheme using the basic element $(\pi)_{\phi},(\pi)_{-\phi}$ ((a)-(c)) and
$(\pi/2)_{\phi}(3\pi/2)_{\phi+\pi}$ ((d)-(e)) as a function of
$\nu^{(0)}_{\textrm{S}}/\nu_{1}$ for $\nu_{1}=65\,$kHz. Using composite pulses
greatly improves the recoupling bandwidth from approximately
$0.1\nu_{1}=6.5\,$kHz to $0.8\nu_{1}=52\,$kHz.

Figure 10: Recoupling intensity $\langle\textrm{S}_{2z}\rangle$ for the
$\mathrm{R}26^{11}_{4}$ pulse scheme obtained from exact calculations and
effective Hamiltonian calculations up to second order as a function of
$\nu^{(0)}_{\textrm{S}}/\nu_{1}$ for $\nu_{1}=65\,$kHz. Subfigures (a)-(c)
correspond to the implementation $\mathrm{R}26^{11}_{4}$ without composite
pulses leading to a recoupling bandwidth of approximately
$0.1\nu_{1}=6.5\,$kHz. For subfigures (d)-(f), $\mathrm{R}26^{11}_{4}$ was
implemented using composite pulses increasing the recoupling bandwidth
significantly to approximately $0.8\nu_{1}=52\,$kHz. The simulation parameters
have been kept consistent with the parameters used for Fig. 8.

##  IV Conclusion

Effective Hamiltonians serve as indispensable tools for comprehending and
developing solid-state NMR experiments. Traditional Floquet theory has proven
particularly efficacious in experiments involving multiple modulation
frequencies, often entailing MAS and employing multiple rf pulses on one or
more spin species.

The prerequisite of a periodic Hamiltonian, inherent in traditional Floquet
theory, can, however, be restrictive. Continuous Floquet theory represents an
extension of its traditional counterpart, eliminating the necessity of a
periodic Hamiltonian. This extension enables the analysis of both non-resonant
and near-resonant experiments, affording a more extensive back-calculation of
the effective Hamiltonian from experimental parameters.

In Continuous Floquet theory, the definition of the effective Hamiltonians
uses a single frequency to describe multiple time dependencies. This feature,
while enhancing the scope of analysis, may complicate the interpretation of
effective Hamiltonians. Furthermore, the definition of the second-order
effective Hamiltonian as a Cauchy principal value introduces certain
inconveniences, potentially leading to decreased computational efficiency. One
disadvantage of Continuous Floquet theory is the fact that the effective
Hamiltonian depends on the length of the sequence unless it is calculated on a
resonance condition where it is the same as the classical effective Floquet
Hamiltonian. This requires a recalculation of the convolution of the Fourier
coefficients with the sinc function if the mixing time is changed.

The effective Hamiltonians in this paper, resolve these shortcomings and
closely align with traditional Floquet theory, ensuring ease and flexibility
in their application to NMR experiments with multiple modulation frequencies.
Notably, Fourier coefficients obtained through analysis within the traditional
Floquet theory framework can be repurposed seamlessly, establishing a unified
foundation for computational methodologies.

In the later part of this manuscript, we focus on applying the presented
framework, emphasizing earlier discussed aspects. Utilizing Fourier
coefficients from literature calculated within traditional Floquet theory, we
investigated the near-resonance behavior of various recoupling pulse schemes.
Our analysis emphasized that the overall duration is the most crucial
parameter for recoupling bandwidth in these experiments.

## Supplementary Material

Simulation of single crystallites using different dipolar coupling strengths
for the discussed pulse schemes, as well as the derivation of the first and
second effective Hamiltonian, can be found in the Supplementary Information.

## Acknowledgments

This research was supported by the ETH Zürich and the Schweizerischer
Nationalfonds zur Förderung der Wissenschaftlichen Forschung (grant nos.
200020_188988 and 200020_219375).

## Author Declarations

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

Conceptualization (MC,ME); Formal analysis (MC); Investigation (MC,ME);
Methodology (MC); Software (MC,ME); Writing – original draft (MC); Writing –
review & editing (MC,ME).

## Data Availability

The data that support the findings of this study are available on
https://doi.org/10.3929/ethz-b-000668159.

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