Zero-Field NMR J-Spectroscopy of Organophosphorus Compounds

Organophosphorus compounds are a wide and diverse class of chemicals playing a crucial role in living organisms. This aspect has been often investigated using nuclear magnetic resonance (NMR), which provides information about molecular structure and function. In this paper, we report the results of theoretical and experimental studies on basic organophosphorus compounds using zero-field NMR, where spin dynamics are investigated in the absence of a magnetic field with the dominant heteronuclear J-coupling. We demonstrate that the zero-field NMR enables distinguishing the chemicals owing to their unique electronic environment even though their spin systems have the same alphabetic designation. Such information can be obtained just in a single measurement, while amplitudes and widths of observed low-field NMR resonances enable the study of processes affecting spin dynamics. An excellent agreement between simulations and measurements of the spectra, particularly in the largest frequency J-couplings range ever reported in zero-field NMR, is demonstrated.

here and apply it for 31 P-1 H systems to further motivate experimental sequence used and described in this study. To be specific, we consider a zero-field NMR experiment with thermal prepolarization, where a sample undergoes sudden transfer from high to ultra-low field. We also assume a very specific detection geometry: since in our setup the NMR tube is positioned above a sensor (the z direction), simultaneously being a magnetometer sensitivity axis, we narrow our analysis only to a detection of the z-component of magnetizationM z . The initial density matrixρ o (neglecting constant non-evolving term) following sudden transfer from high to zero field given by:  Figure S1: Time dependence of the static magnetic field during sudden transition experiments without (a) and with (b) an additional DC magnetic pulse. The initial period of prepolarization in the strong field B pol is followed by a non-adiabatic transport through the field of guiding field (eventually the sample reaches a magnetically-shielded region). The guiding field B sol is then switched off suddenly and the signal is either observed straight away (a) or a sharp DC pulse is applied in y and enhanced signal is subsequently observed (b).
whereÎ z andŜ z are operators of the z-component of the total spin of phosphorus and hydrogen groups respectively, and γ I and γ S are the corresponding gyromagnetic ratios.
This density matrix represents a high-field state which was preserved (neglecting relaxation) during a sudden transfer to zero-field. Since the density matrix of this high-field state does not commute with the zero-field Hamiltonian, determined by the J-coupling,Ĥ J = J PHÎ ·Ŝ, whereŜ andÎ are the total spin operators of 31 P and 1 H groups, the state immediately starts to evolve giving rise to a detectable signal (Fig. S1a). In a complementary approach, after switching off the guiding field, a sharp, transverse DC pulse with a duration τ p and a magnetic-field strength B p is applied (Fig. S1b). Neglecting the J-coupling evolution during the pulse, the density matrix (in which we neglect the xy terms not relevant in our detection geometry) after the pulse is given aŝ where θ I = B p τ p γ I and θ S = B p τ p γ S are rotation angles of spin I and S due to the pulse.
In both cases considered here [Eqs. (S1) and (S2)], the relevant part of the initial density matrix is a mixture of two terms: one proportional toÎ z +Ŝ z , which remains stationary S2 under zero-field Hamiltonian and corresponds to decaying "static" magnetization and the other proportional toÎ z −Ŝ z , which evolves giving rise to an oscillating signal. Since in a typical NMR experiment an oscillating signal is detected, the term proportional toÎ z −Ŝ z should be maximized. For two spins with same sign of gyromagnetic ratio (in particular 1 H and 31 P), a pulse angle maximizingÎ z −Ŝ z should satisfy conditions: cos θ I = 1, cos θ S = −1 or cos θ I = −1, cos θ S = 1. For the 31 P-1 H system, where γ P /γ H ≈ 2/5, this conditions is achieved in particular when θ H ≈ 5π, θ P ≈ 2π.
To compare signal amplitudes after sudden transfer, we turn to calculate the detectable component of magnetization M z . For simplicity, we will only consider oscillating terms in initial density matrices, while the pulse length is picked to maximize theÎ z −Ŝ z term. In that case, an oscillating magnetization for both arrangements in proportional to: By comparing the z-magnetization amplitudes for sudden transfer without [Eq. (S3)] and with the pulse[Eq. (S4)], one can see that the application of the pulse changes signal amplitude. In particular, for the 31 P-1 H system the maximal signal enhancement is To support our considerations, we simulate zero-field NMR signals of the 31 P-1 H system using the SpinDynamica package for Mathematica. 4 The calculations are performed using a finite-length pulse with an amplitude of 50-µT, which closely matches the experimental conditions (J-coupling evolution is present during the pulse). The amplitudes of such signals for different pulse lengths are shown as red points in Fig. S2a. The signal-pulse dependence is then compared with a more general analytical formula (solid black line in Fig. S2a): derived in Ref. 3 for a similar 13 C-1 H spin system. Figure S2b shows simulated spectra for a simple 31 P-1 H system following sudden transition to zero-field with and without the optimal pulsing, while the experimental signal enhancement using a transverse DC pulse is shown for trimethyl phosphate in Fig. S2c.  Figure S2: (a) Amplitude of the zero-field NMR signal of the 31 P-1 H nuclear system versus the 1 H rotation angle θ H induced by the field applied in y. The red dots are results of numerical simulations (SpinDynamica), while the black line corresponds to the signal amplitude determined based on Eq. (S6). The signal is normalized to the maximum amplitude. (b) Simulated J-spectra of the 31 P-1 H spin system with J PH = 11 Hz following sudden transition to zero-field with/without the optimum y pulse. (c) Experimental J-spectra of trimethyl phosphate after the sudden transition to zero-field with/without optimum y pulse. The spectra are the result of 32 averaged transients.

Analysis of (XA n )B m spin systems via first-order perturbation theory
Here we show detailed results of the first-order perturbative calculation for dichlorvos and dimethyl phosphite spin systems. The values of all observable transitions are presented in Tables S1 and S2. Note that not all of the transitions in the case of dichlorvos are visible in experimental spectra (see main text). Table S1: Frequencies (ν) of allowed transitions between levels characterized with the quantum numbers F and F T , corresponding to the dominant spin subsystem XA 6 and total spin of the whole (XA 6 )B system, obtained using first-order perturbation theory. The J-coupling constants used for perturbative calculation are J A PH = 11.4 Hz, J B PH = 5.2 Hz.