A physical interpretation of the Floquet description of magic angle spinning nuclear magnetic resonance spectroscopy

Theoretical description of pulse induced resonances in the homonuclear PIRATE experiment

Rotor-synchronous π pulses applied to protons (S) enhance homonuclear polarisation transfer between two spins (I) such as ¹³C or ¹⁵N as long as at least a single I-S heteronuclear dipolar-coupling interaction exists. The enhancement is maximum when the chemical-shift difference Δν between two spins equals an integer multiple, n, of the pulse-modulation frequency, which is half the rotor frequency ν_r. This condition, applied in the Pulse Induced Resonance with Angular dependent Total Enhancement (PIRATE) experiment, can be generalised for any spacing of the pulses k/ν_r such that Δν=nν_r2k . We show, using average Hamiltonian theory (AHT) and Floquet theory, that the resonance conditions promote a second-order recoupling consisting of a cross-term between the homonuclear and heteronuclear dipolar interactions in a three-spin system. The minimum requirement is a coupling between the two I spins and a coupling of one of the I spins to the S spin. The effective Hamiltonian at the resonance conditions contains three-spin operators of the form 2I₁^±I₂^∓S_z with a non-zero effective dipolar coupling. Theoretical analysis shows that the effective strength of the resonance conditions decreases with increasing values of k and n. The theory is backed by numerical simulations, and experimental results on fully labelled ¹³C-glycine demonstrating the efficiency of the different resonance condition for k=1,2 at various spinning frequencies.

A continuous approach to Floquet theory for pulse-sequence optimization in solid-state NMR

We present a framework that uses a continuous frequency space to describe and design solid-state nuclear magnetic resonance (NMR) experiments. The approach is similar to the well-established Floquet treatment for NMR, but it is not restricted to periodic Hamiltonians and allows the design of experiments in a reverse fashion. The framework is based on perturbation theory on a continuous Fourier space, which leads to effective, i.e., time-independent, Hamiltonians. It allows the back-calculation of the pulse scheme from the desired effective Hamiltonian as a function of spin-system parameters. We show as an example how to back-calculate the rf irradiation in the MIRROR experiment from the desired chemical-shift offset behavior of the sequence.

Floquet theory in magnetic resonance: Formalism and applications

Floquet theory is an elegant mathematical formalism originally developed to solve time-dependent differential equations. Besides other fields, it has found applications in optical spectroscopy and nuclear magnetic resonance (NMR). This review attempts to give a perspective of the Floquet formalism as applied in NMR and shows how it allows one to solve various problems with a focus on solid-state NMR. We include both matrix- and operator-based approaches. We discuss different problems where the Hamiltonian changes with time in a periodic way. Such situations occur, for example, in solid-state NMR experiments where the time dependence of the Hamiltonian originates either from magic-angle spinning or from the application of amplitude- or phase-modulated radiofrequency fields, or from both. Specific cases include multiple-quantum and multiple-frequency excitation schemes. In all these cases, Floquet analysis allows one to define an effective Hamiltonian and, moreover, to treat cases that cannot be described by the more popularly used and simpler-looking average Hamiltonian theory based on the Magnus expansion. An important example is given by spin dynamics originating from multiple-quantum phenomena (level crossings). We show that the Floquet formalism is a very general approach for solving diverse problems in spectroscopy.

Photoinduced Electron Transfer in Donor-Acceptor Complexes: Isotope Effect and Dynamic Symmetry Breaking

Electron-nuclear (vibronic) coupling has emerged as an important factor in determining the efficiency of energy transfer and charge separation in natural and artificial photosynthetic systems. Here we investigate the photoinduced charge-transfer process in a hydrogen-bonded donor-acceptor molecular complex. By using real-time quantum-classical simulations based on time-dependent Kohn-Sham equations, we follow in detail the relaxation from the Franck-Condon point to the region of strong nonadiabatic coupling where electron transfer occurs. We elucidate how the charge transfer is coupled to specific vibrational modes and how it is affected by isotope substitution. The importance of resonance in nuclear and electron dynamics and the role of dynamic symmetry breaking are emphasized. Using the dipole moment as a descriptive parameter, exchange of angular momentum between nuclear and electronic subsystems in an electron-nuclear resonant process is inferred. The performed simulations support a nonadiabatic conversion via adiabatic passage process that was recently put forward. These results are relevant in deriving rational design principles for solar-to-fuel conversion devices.

Cross-Effect Dynamic Nuclear Polarization Explained: Polarization, Depolarization, and Oversaturation

The scope of this Perspective is to analytically describe NMR hyperpolarization by the three-spin cross effect (CE) dynamic nuclear polarization (DNP) using an effective Hamiltonian concept. We apply, for the first time, the bimodal operator-based Floquet theory in the Zeeman-interaction frame for two and three coupled spins to derive the known interaction Hamiltonian for CE-DNP. With a unified understanding of CE-DNP, and supported by empirical observation of the state of electron spin polarization under the given experimental conditions, we explain diverse manifestations of CE from oversaturation, enhanced hyperpolarization by broad-band saturation, to nuclear spin depolarization under magic-angle spinning.

Significance of symmetry in the nuclear spin Hamiltonian for efficient heteronuclear dipolar decoupling in solid-state NMR: A Floquet description of supercycled rCW schemes

Symmetry plays an important role in the retention or annihilation of a desired interaction Hamiltonian in NMR experiments. Here, we explore the role of symmetry in the radio-frequency interaction frame Hamiltonian of the refocused-continuous-wave (rCW) pulse scheme that leads to efficient ¹H heteronuclear decoupling in solid-state NMR. It is demonstrated that anti-periodic symmetry of single-spin operators (I_x, I_y, I_z) in the interaction frame can lead to complete annihilation of the ¹H-¹H homonuclear dipolar coupling effects that induce line broadening in solid-state NMR experiments. This symmetry also plays a critical role in cancelling or minimizing the effect of ¹H chemical-shift anisotropy in the effective Hamiltonian. An analytical description based on Floquet theory is presented here along with experimental evidences to understand the decoupling efficiency of supercycled (concatenated) rCW scheme.

Five decades of homonuclear dipolar decoupling in solid-state NMR: Status and outlook

It has been slightly more than fifty years since the first homonuclear spin decoupling scheme, Lee-Goldburg decoupling, was proposed for removing homonuclear dipolar interactions in solid-state nuclear magnetic resonance. A family of such schemes has made observation of high-resolution NMR spectra of abundant spins possible in various applications in solid state. This review outlines the strategies used in this field and the future prospects of homonuclear spin decoupling in solid-state NMR.

Understanding cross-polarization (CP) NMR experiments through dipolar truncation

A theoretical model based on the phenomenon of dipolar truncation is proposed to explain the nuances of polarization transfer from abundant to less-abundant nuclei in cross-polarization (CP) NMR experiments. Specifically, the transfer of polarization from protons to carbons (in solids) in strongly coupled systems is described in terms of effective Hamiltonians based on dipolar truncation. Through suitable model spin systems, the important role of dipolar truncation in the propagation of spin polarization in CP experiments is outlined. We believe that the analytic theory presented herein provides a convenient framework for modeling polarization transfer in strongly coupled systems.

Derivation of a Floquet formalism within a natural framework

Many biological systems experience a periodic environment. Floquet theory is a mathematical tool to deal with such time periodic systems. It is not often applied in biology, because linkage between the mathematics and the biology is not available. To create this linkage, we derive the Floquet theory for natural systems. We construct a framework, where the rotation of the Earth is causing the periodicity. Within this framework the angular momentum operator is introduced to describe the Earth’s rotation. The Fourier operators and the Fourier states are defined to link the rotation to the biological system. Using these operators, the biological system can be transformed into a rotating frame in which the environment becomes static. In this rotating frame the Floquet solution can be derived. Two examples demonstrate how to apply this natural framework.

Sweep direction and efficiency of the swept-frequency two pulse phase modulated scheme for heteronuclear dipolar-decoupling in solid-state NMR

We present here a bimodal Floquet theoretical and experimental investigation of the direction of sweep in the swept-frequency two pulse phase modulated (SW(f)-TPPM) scheme used for heteronuclear dipolar decoupling in solid-state NMR. The efficiency of the decoupling turns out to be independent of the sweep direction.

1H Homonuclear dipolar decoupling using symmetry-based pulse sequences at ultra fast magic-angle spinning frequencies

We demonstrate here the application of symmetry-based pulse sequences for homonuclear dipolar decoupling in solid-state NMR at magic-angle spinning (MAS) frequencies up to 65 kHz using moderate radiofrequency (RF) amplitudes. Theoretical arguments favouring the requirement of low RF amplitudes at high MAS frequencies are given for these sequences. A comparison with wPMLGmmxx¯ is given at 65 kHz of MAS frequency to emphasise that the symmetry-based pulse sequences have a lower RF amplitude requirement at high MAS frequencies.

Why does PMLG proton decoupling work at 65kHz MAS?

Schemes such as phase-modulated Lee-Goldburg (PMLG) for homonuclear dipolar decoupling have been shown to yield high-resolution (1)H spectra at high magic-angle spinning (MAS) frequencies of 50-70kHz. This is at variance to the commonly held notion that these methods require MAS frequencies not comparable to the cycle frequencies of the pulse schemes. Here, a theoretical argument, based on bimodal Floquet theory, is presented to explain this aspect together with conditions where PMLG type of schemes may be successful at high MAS frequencies.

Floquet-van Vleck analysis of heteronuclear spin decoupling in solids: the effect of spinning and decoupling sidebands on the spectrum

We investigate the effect of magic angle spinning on heteronuclear spin decoupling in solids. We use an analytical Floquet-van Vleck formalism to derive expressions for the powder-averaged signal as a function of time. These expressions show that the spectrum consists of a centerband at the isotropic frequency of the observed spin, omega(0), and rotational decoupling sidebands at omega(0)+/-omega(1)+/-momega(r), where omega(1) is the decoupling field strength and omega(r) is the rotation frequency. Rotary resonance occurs when the rotational decoupling sidebands overlap with the centerband. Away from the rotary resonance conditions, the intensity of the centerband as a function of omega(r)/omega(1) is simply related to the total intensity of the rotational decoupling sidebands. Notably, in the absence of offset terms it is shown that as omega(1) decreases, the centerband intensity can decrease without any associated broadening. Furthermore, the centerband width is shown to be independent of spinning speed, to second order for the effects we consider. The effects of I spin chemical shift anisotropy and homonuclear dipolar couplings are also investigated. The analytical results are compared to simulations and experiments.

Coherent cross-polarization theory for a spin-12 coupled to a general object

Zero-order average-Hamiltonian theory is used to extend the product-operator description of coherent spin-spin cross-polarization to the case of a spin-12 coupled to a general object, like a molecular rotor or a quantum oscillator. The object, which is not necessarily in a Boltzmann equilibrium state, is assumed to have no interaction with the lattice and no internal relaxation capacity. The Bloch-Wangsness-Redfield (BWR) theory for incoherent processes like spin-lattice relaxation does not apply for such an isolated spin-object pair. Nevertheless spectral density at the Larmor frequency, of key importance in BWR theory, also plays a central role in object-induced spin polarization. Spectral density in our theory is represented by quantum operators J(-) and J(+). If J(-) and J(+) do not commute, the spin-object coupling may cause spin polarization in an initially saturated spin system. This represents a coherent mechanism for spin cooling, which in specific cases may lead to enhanced spin polarization above the thermal equilibrium value. A master equation is derived for general spin-object cross-polarization, and applied to the case of a spin pair inside a uniaxial rotor, and a spin coupled to a microelectronic LC circuit. Copyright 2000 Academic Press.

Anomalous rotational resonance spectra in magic-angle spinning NMR

Magic-angle spinning NMR spectra of samples containing dilute spin-1/2 pairs display broadenings or splittings when a rotational resonance condition is satisfied, meaning that a small integer multiple of the spinning frequency matches the difference in the two isotropic shift frequencies. We show experimental rotational resonance NMR spectra of a 13C2-labeled retinal which are in qualitative disagreement with existing theory. We propose an explanation of these anomalous rotational spectra involving residual heteronuclear couplings between the 13C nuclei and the neighboring 1H nuclei. These couplings strongly influence the rotational resonance 13C spectrum, despite the presence of a strong radiofrequency decoupling field at the 1H Larmor frequency. We model the residual heteronuclear couplings by differential transverse relaxation of the 13C single-quantum coherences. We present a superoperator theory of the phenomenon and describe a numerical algorithm for rapid Liouville space simulations in periodic systems. Good agreement with experimental results is obtained by using a biexponential transverse relaxation model for each spin site.