Forward Linear Prediction in the Indirect Dimension of 2D Data

Forward linear prediction in 1D data is the calculation of new data points after the end of the acquisition time based on the observed data points. It can be used to artificially increase the acquisition time and thereby the spectral resolution. Forward linear prediction can also be used in the indirect dimension (F1) of 2D data to artificially increase the number of slices collected in the experiment and thereby improve the spectral resolution in the F1 domain. This represents a huge time saving as fewer slices need be acquired. The figure below shows the HMQC spectrum of 3-heptanone. The data in the top panel was collected with 256 slices in F1 and took 12.8 minutes to acquire. The data in the center panel was collected with 40 slices in F1 and took only 2.0 minutes to acquire. It has much lower resolution in F1 than the data in the top panel. The data in the bottom panel was produced from the same raw data as the spectrum in the center panel except forward linear prediction was applied in the F1 domain. It has comparable F1 resolution to that in the top panel but took less than 16% of the time to collect!

2D NMR

Some students treat 2D NMR like a mysterious black box. Making the connection between how 1D NMR and 2D NMR experiments are conducted is not really that complicated. I hope that this explanation helps.

A 2D pulse sequence is simply a 1D pulse sequence with an incremented delay. In fact, one can think of a T₁ measurement as a simple 2D sequence (where in this case only one set of Fourier transforms is calculated). A 2D pulse sequence can be divided into 3 periods; preparation, evolution and detection as can be seen in the figure below.These periods can consist of a series of pulses, delays and/or gradients suitable to the property of the spijn system being measured. In 2D NMR, a series of FIDs are collected as a function of the evolution time, t1, to get signals, S(t1,t2). Each of these FIDs is Fourier transformed with respect to t2 to give a set of 1D NMR spectra, S(t1,F2). This series of spectra is encoded with information about what happened during the evolution period. The information can be decoded by doing a second Fourier transform with respect to the evolution period, t1. Corresponding points in the NMR spectra, S(t1,F2) are plotted against t1. The resulting interferograms are Fourier transformed to give a two dimensional stacked plot or contour map, S(F1,F2). This is the 2D spectrum.

What is the Difference Between a Broadband and an Inverse Broadband Probe?

I am sometimes asked, what is the difference between a broadband probe and an inverse broadband NMR probe? Both probes typically have a fixed proton channel, a tunable broadband channel and a fixed ²H lock channel. The difference lies in the coil positions with respect to the sample. Broadband probes are optimized to observe an X nucleus (typically all nuclei with frequencies between ¹⁵N and ³¹P). The broadband, X coil is positioned closest to the sample to maximize the filling factor and hence the X nucleus sensitivity. The ¹H coil is outside of the X coil and can be used for either ¹H observation or ¹H decoupling. The filling factor for this coil is lower than that of the X coil. In an inverse broadband probe, the ¹H coil is closest to the sample maximizing the ¹H sensitivity. The X coil is outside of the ¹H coil and can be used for either X nucleus observation or decoupling. The X nucleus in an inverse broadband probe has much less sensitivity than in a broadband probe. The choice of probe depends entirely on whether the user desires more ¹H or more X sensitivity. For example, The ¹³C spectra of dilute samples should be run in a broadband probe whereas ¹H - ¹³C or ¹H - ¹⁵N HSQC spectra of dilute samples should be run in an inverse broadband probe.

1H NMR Spectra of Solids

Students will sometimes come into the lab with an insoluble organic material and ask me for a ¹H NMR spectrum of the solid. My response is usually, "I'll draw it for you." after which I will draw a broad lump on the nearest piece of scrap paper and hand it to them with a smile. I will then explain why.

In the liquid state, molecules are randomly tumbling and the dipole-dipole coupling between the protons is averaged to zero allowing the observation of sharp resonances, isotropic chemical shifts and J coupling. In solids, the molecules are more rigid and each proton is dipolar coupled to many other protons. All of the protons are coupled together in a large network. Since the dipolar coupling is much larger than the chemical shift range and any J coupling, the spectrum appears as a broad lump, typically about 50 kHz or so in width. Magic angle spinning will help the situation but is not usually fast enough to average the dipolar coupling. "High" resolution ¹H spectra of solids can be obtained (with great effort!) by combining magic angle spinning and multiple pulse decoupling (CRAMPS). In this method points are sampled during windows in the multiple pulse decoupling scheme. Typical resolution in a CRAMPS spectrum is still only about 1 ppm or so. In the figure below, ¹H NMR spectra are compared for a static organic solid, an organic solid spinning at the magic angle at 12 kHz and the same solid dissolved in a solvent. In the MAS spectrum one can see that the ¹H resonance is narrower with spinning sidebands at 12 kHz, however the resolution is insufficient to observe discreet resonances.

Solution vs Solid State MAS NMR

The number of peaks present in a standard ¹³C solution state NMR spectrum with proton decoupling depends on the number of symmetry independant carbon atoms present in the molecule. If the molecule has no symmetry then the number of peaks equals the number of different types of carbon in the molecule. When the molecule has symmetry, the number of ¹³C resonances is reduced by the symmetry. The molecular symmetry determines the number of resonances. In the solid state, on the other hand, the molecules are part of a crystal lattice. In solids it is the crystallographic symmetry rather than the molecular symmetry that determines the number of peaks in the spectrum. There will be a resonance for each carbon in the asymmetric unit of the crystal. This is illustrated in the figure below. In this case, for the solution state spectrum, there is half as many ¹³C resonances as there are carbon atoms due to the molecular symmetry. In the solid, there is one entire molecule in the asymmetric unit of the crystal therefore the ¹³C CPMAS spectrum has one peak for each carbon in the molecule.

Gradient Calibration - 1D MRI

When people think about magnetic resonance imaging (MRI), they often think about the huge whole body imagers in hospitals. NMR spectroscopists use one dimensional MRI in a specially prepared sample to calibrate the Z gradient strength of their spectrometers. The sample consists of a plastic disk with a precisely known thickness immersed in a column of water (see the figure below). 1D MRI is conceptually quite simple - a linear field gradient is applied during the collection of an FID. Since the magnetic field varies across the sample and the NMR frequency is proportional to the magnetic field strength, the resulting NMR spectrum represents a one dimensional image of the sample. The spectrum has a "notch" missing as a result of the plastic disk. The width of the "notch" is proportional to the applied gradient strength and the thickness of the plastic disk. If the thickness of the plastic disk is precisely known, then the strength of the applied gradient can be calculated. The spectrum is rounded at the edges as a result of the gradient strength falling off away from the center of the sample. A modified version of this experiment using an echo is used as a routine method of calibrating gradient strengths.

Forward Linear Prediction

The digital resolution in an NMR spectrum can be improved by zero filling the FID (i.e. adding zeros to the end of the FID before Fourier transformation). Forward linear prediction, on the other hand, can be used to improve both the digital and real resolution in a spectrum. Forward linear prediction uses the data collected in an FID to predict data after the receiver was turned off. Both processing techniques artificially increase the acquisition time, however it is only forward linear prediction which adds new information to the spectrum. In the figure below a truncated FID is transformed untreated, with zero filling and with forward linear prediction.

Is My 1H NMR Spectrum Quantitative?

I often have students ask me, "Is my ¹H NMR spectrum quantitative?". Of course we all know that the area under the signals of properly recorded ¹H NMR spectra represent the number of protons responsible for the signal, however the answer to the question may be a bit more involved than a simple "yes". The following things must be taken into account.
Relaxation - In order for a spectrum to be quantitative, each of the NMR signals must be sufficiently relaxed to equilibrium before a pulse is applied. One should ideally have an idea what the T₁ is for each of the signals and make sure that the interpulse delay (i.e. the sum of the acquisition time and the relaxation delay) is at least 5 times greater than the longest T₁ if 90 degree pulses are used or at least twice the longest T₁ if 30 degree pulses are used. This should provide data quantitative to about 95%. If more precise data is required, for example in kinetic isotope studies, the interpulse delay must be made even longer.
Baseline Roll - The reliability of integrals depends on the quality of the baseline in the spectrum. One should ensure a perfectly flat baseline is achieved by way of baseline correction routines.
Precise Phasing - Phasing errors in the spectrum (even minor ones) will result in imprecise integrals. Automatic phasing routines cannot always be relied upon. One should expand each signal vertically to ensure a proper absorptive phase using manual baseline correction.
Signal-to-Noise Ratio - Noisy spectra will give noisy integrals. The noise is directly related to the precision of the integral. The concentration of the sample or the number of scans should be chosen so that the signal-to-noise ratio is at least 100:1.
Excitation Profile - The pulses used to measure the spectrum must be sufficiently short to produce a flat excitation profile over the entire spectrum. This is potentially a problem for paramagnetic samples whose resonances span a very large chemical shift range. For more on excitation profiles click here.
Chemical Exchange - The signals from exchangeable protons may be very broad (therefore low signal-to-noise ratio) and will give imprecise integrals. If a deuterated solvent with exchangeable protons is used, then the exchangeable protons in the sample will exchange with the exchangeable deuterons of the solvent thus reducing or eliminating the exchangeable proton signals of the sample. This can also be used to your advantage as an assignment tool.
Solvent Suppression by Presaturation - When a solvent line is presaturated, the integrals of the signals close to the solvent signal will be diminished. If water suppression is used, some exchangeable signals from the sample could also be diminished due to exchange from the saturated protons of the solvent.
Edge Effects - Some NMR setups will produce baseline artifacts (smiles or frowns) at the edges of the observation window. One should choose a spectral width sufficiently large so that the signals will not be affected by this artifact.

Probe Coil Geometry

The NMR sample sits inside a coil within the NMR probe. The coil is part of a tuned circuit which delivers radio frequency (RF) pulses to the sample and detects the RF signals from the sample. The RF fields delivered by and detected from the coil must be perpendicular to the static magnetic field, Bo. The most efficient way to do this is with a horizontal solenoid coil. Such coils are commonly used in NMR probes for solids. The probe must be removed from the magnet when samples are changed. A less efficient, but more convenient, Helmholtz coil is commonly used for liquids samples. Like horizontal solenoid coils, vertical Helmholtz coils deliver and detect RF fields perpendicular to Bo, but due to their geometry they allow samples to be inserted and ejected form the probe with a pneumatic system. The probe need not be removed from the magnet to change samples.

The Importance of a Proper Hartmann-Hahn Match in CPMAS NMR

Cross polarization (CP) occurs when the Hartmann -Hahn condition is met by way of the dipolar coupling between the abundant isotope and the dilute isotope. The Hartmann -Hahn match is very sensitive to power levels when the dipolar coupling between the abundant and dilute isotopes is comparable to the MAS spinning speed. The matching curve in this case is separated into a series of minima and maxima. In some materials, the dipolar coupling between the dilute nuclei and the protons can be very dependent on the chemical environment of the dilute nucleus. In such materials, the appearance of the spectrum is very dependent on the quality of the Hartmann-Hahn match. Differences in the spectra as a function of the Hartmann-Hahn matching condition can be used to learn something about the heteronuclear dipolar coupling for different chemical sites. In the figure below, the ²⁹Si CPMAS spectrum is shown for the mineral sepiolite with and without a proper Hartmann-Hahn match. There are three resolved sites with intensity ratio 1:1:1. Once can see that when the Hartmann-Hahn condition is slightly off, the intensity of one of the sites is drastically reduced with respect to the other two.

1H Chemical Shift Referencing in Aqueous Solutions

Typically one uses either dissolved TMS or the residual protons of commercially available deuterated solvents as convenient chemical shift references for solutions in organic solvents. This is not possible for aqueous solutions as TMS is not soluble in water and the 1H chemical shift of water is notoriously dependant on temperature, pH, chemical exchange etc... A common replacement for TMS for aqueous solutions is sodium 2,2-dimethyl-2-sila-pentane-5-sulfonate (DSS). The chemical shift of the three equivalent methyl groups of this compound can be set at 0 ppm.