²H MAS NMR of Methionine

The following is a least-squares fitting example of a \(^{2}\text{H}\) MAS NMR spectrum of Methionine. The experimental dataset is a part of DMFIT [1] examples. We thank Dr. Dominique Massiot for sharing the dataset.

import csdmpy as cp
import matplotlib.pyplot as plt
from lmfit import Minimizer

from mrsimulator import Simulator, SpinSystem, Site
from mrsimulator.method.lib import BlochDecaySpectrum
from mrsimulator import signal_processor as sp
from mrsimulator.utils import spectral_fitting as sf
from mrsimulator.utils import get_spectral_dimensions
from mrsimulator.spin_system.tensors import SymmetricTensor

Import the dataset

host = "https://nmr.cemhti.cnrs-orleans.fr/Dmfit/Help/csdm/"
filename = "2H methiodine MAS.csdf"
experiment = cp.load(host + filename)

# standard deviation of noise from the dataset
sigma = 0.3026282

# For spectral fitting, we only focus on the real part of the complex dataset
experiment = experiment.real

# Convert the coordinates along each dimension from Hz to ppm.
_ = [item.to("ppm", "nmr_frequency_ratio") for item in experiment.dimensions]

# plot of the dataset.
plt.figure(figsize=(4.25, 3.0))
ax = plt.subplot(projection="csdm")
ax.plot(experiment, color="black", linewidth=0.5, label="Experiment")
ax.set_xlim(600, -700)
plt.grid()
plt.tight_layout()
plt.show()

Create a fitting model

Spin System

H_2 = Site(
    isotope="2H",
    isotropic_chemical_shift=-57.12,  # in ppm,
    quadrupolar=SymmetricTensor(Cq=3e4, eta=0.0),  # Cq in Hz
)

spin_systems = [SpinSystem(sites=[H_2])]

Method

# Get the spectral dimension parameters from the experiment.
spectral_dims = get_spectral_dimensions(experiment)

MAS = BlochDecaySpectrum(
    channels=["2H"],
    magnetic_flux_density=9.395,  # in T
    rotor_frequency=4517.1,  # in Hz
    spectral_dimensions=spectral_dims,
    experiment=experiment,  # experimental dataset
)

# Optimize the script by pre-setting the transition pathways for each spin system from
# the method.
for sys in spin_systems:
    sys.transition_pathways = MAS.get_transition_pathways(sys)

Guess Model Spectrum

# Simulation
# ----------
sim = Simulator(spin_systems=spin_systems, methods=[MAS])
sim.run()

# Post Simulation Processing
# --------------------------
processor = sp.SignalProcessor(
    operations=[
        sp.IFFT(),
        sp.apodization.Exponential(FWHM="60 Hz"),
        sp.FFT(),
        sp.Scale(factor=140),
    ]
)
processed_dataset = processor.apply_operations(dataset=sim.methods[0].simulation).real

# Plot of the guess Spectrum
# --------------------------
plt.figure(figsize=(4.25, 3.0))
ax = plt.subplot(projection="csdm")
ax.plot(experiment, color="black", linewidth=0.5, label="Experiment")
ax.plot(processed_dataset, linewidth=2, alpha=0.6, label="Guess Spectrum")
ax.set_xlim(600, -700)
plt.grid()
plt.legend()
plt.tight_layout()
plt.show()

Least-squares minimization with LMFIT

Use the make_LMFIT_params() for a quick setup of the fitting parameters.

params = sf.make_LMFIT_params(sim, processor)
params["sys_0_site_0_isotropic_chemical_shift"].vary = False
print(params.pretty_print(columns=["value", "min", "max", "vary", "expr"]))

Out:

Name                                      Value      Min      Max     Vary     Expr
SP_0_operation_1_Exponential_FWHM            60     -inf      inf     True     None
SP_0_operation_3_Scale_factor               140     -inf      inf     True     None
sys_0_abundance                             100        0      100    False      100
sys_0_site_0_isotropic_chemical_shift    -57.12     -inf      inf    False     None
sys_0_site_0_quadrupolar_Cq               3e+04     -inf      inf     True     None
sys_0_site_0_quadrupolar_eta                  0        0        1     True     None
None

Solve the minimizer using LMFIT

minner = Minimizer(sf.LMFIT_min_function, params, fcn_args=(sim, processor, sigma))
result = minner.minimize()
result

Fit Statistics

fitting method	leastsq
# function evals	21
# data points	8192
# variables	4
chi-square	1350041.14
reduced chi-square	164.880452
Akaike info crit.	41825.9668
Bayesian info crit.	41854.0105

Variables

name	value	standard error	relative error	initial value	min	max	vary	expression
sys_0_site_0_isotropic_chemical_shift	-57.1200000	0.00000000	(0.00%)	-57.12	-inf	inf	False
sys_0_site_0_quadrupolar_Cq	33652.7963	79.9919768	(0.24%)	30000.0	-inf	inf	True
sys_0_site_0_quadrupolar_eta	0.32118873	0.00436216	(1.36%)	0.0	0.00000000	1.00000000	True
sys_0_abundance	100.000000	0.00000000	(0.00%)	100	0.00000000	100.000000	False	100
SP_0_operation_1_Exponential_FWHM	62.9584689	0.32076792	(0.51%)	60.0	-inf	inf	True
SP_0_operation_3_Scale_factor	132.896559	0.50526319	(0.38%)	140.0	-inf	inf	True

Correlations (unreported correlations are < 0.100)

SP_0_operation_1_Exponential_FWHM	SP_0_operation_3_Scale_factor	0.6665
sys_0_site_0_quadrupolar_Cq	sys_0_site_0_quadrupolar_eta	-0.2857
sys_0_site_0_quadrupolar_Cq	SP_0_operation_3_Scale_factor	0.2601
sys_0_site_0_quadrupolar_eta	SP_0_operation_3_Scale_factor	0.1075

The best fit solution

best_fit = sf.bestfit(sim, processor)[0].real
residuals = sf.residuals(sim, processor)[0].real

# Plot the spectrum
plt.figure(figsize=(4.25, 3.0))
ax = plt.subplot(projection="csdm")
ax.plot(experiment, color="black", linewidth=0.5, label="Experiment")
ax.plot(residuals, color="gray", linewidth=0.5, label="Residual")
ax.plot(best_fit, linewidth=2, alpha=0.6, label="Best Fit")
ax.set_xlim(600, -700)
plt.grid()
plt.legend()
plt.tight_layout()
plt.show()

[1]

D.Massiot, F.Fayon, M.Capron, I.King, S.Le Calvé, B.Alonso, J.O.Durand, B.Bujoli, Z.Gan, G.Hoatson, ‘Modelling one and two-dimensional solid-state NMR spectra.’, Magn. Reson. Chem. 40 70-76 (2002) DOI: 10.1002/mrc.984