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General Principles of Two Dimensional (2D) NMR
In NMR the most useful information comes from the interactions between two nuclei,
either through the bonds which connect them (J-coupling interaction) or directly through space
(NOE interaction). One can look at these interactions one at a time by irradiating one resonance
in the proton spectrum (either during the relaxation delay or during acquisition) and looking at
the effect on the intensity or coupling pattern of another resonance. 2D NMR essentially allows
us to irradiate all of the chemical shifts in one experiment and gives us a matrix or twodimensional
map of all of the affected nuclei. There are four steps to any 2D experiment:
1. Preparation: Excite nucleus A, creating magnetization in the x-y plane
2. Evolution: Measure the chemical shift of nucleus A.
3. Mixing: Transfer magnetization from nucleus A to nucleus B (via J or NOE).
4. Detection: Measure the chemical shift of nucleus B.
Of course, all possible pairs of nuclei in the sample go through this process at the same time.
Preparation is usually just a 90o pulse which excites all of the sample nuclei simultaneously.
Detection is simply recording an FID and finding the frequency of nucleus B by Fourier
transformation. To get a second dimension, we have to measure the chemical shift of nucleus A
before it passes its magnetization to nucleus B. This is accomplished by simply waiting a period
of time (called t1, the evolution period) and letting the nucleus A magnetization rotate in the x-y
plane. The experiment is repeated many times over (for example, 512 times), recording the FID
each time with the delay time t1 incremented by a fixed amount. The time course of the nucleus
A magnetization as a function of t1 (determined by its effect on the final FID) is used to define
how fast it rotates and thus its chemical shift. Mixing is a combination of RF pulses and/or delay
periods which induce the magnetization to jump from A to B as a result of either a J coupling or
an NOE interaction (close proximity in space). Different 2D experiments (e.g., NOESY, COSY,
HETCOR, etc.) differ primarily in the mixing sequence, since in each one we are trying to define
the relationship between A and B within the molecule in a different way.
The raw data from a 2D experiment consist of a series of FIDs, each acquired with a
slightly longer t1 delay than the previous one. Varian creates an array of FIDs, with the t1 delay
(parameter d2) arrayed. The first step in processing a 2D dataset is to Fourier transform each of
the FIDs in the array. The resulting spectra are loaded into a matrix with the rows representing
individual spectra in order of t1 value, with the smallest t1 value as the bottom row (Figure 1,
next page). The horizontal axis is labelled F2, which is the chemical shift observed directly in
each FID, and the vertical axis is t1, the evolution delay. Each row in the 2D matrix represents a
spectrum acquired with a different t1 delay, and each column in the matrix represents either noise
(if the F2 value of that column is in a noise region of the spectrum) or, if F2 = FB, the column is a
t1 "FID" with maximum intensity at the bottom and oscillating in a decaying fashion as we move
up to higher t1 values (Figure 1). The frequency of this oscillation is just the chemical shift of
nucleus A. Of course, a real sample has more than one peak in its spectrum, so there would be
other columns containing different t1 FIDs.
The second step in processing the 2D data is to perform a second Fourier transform on
each of the columns of the matrix. Most of columns will represent noise, but when we reach a
column which falls on an F2 peak, transformation of the t1 FID gives a spectrum in F1, with a peak
at the chemical shift of nucleus A (Figure 2). The final 2D spectrum is a matrix of numerical
values which has a pocket of intensity at the intersection of the horizontal line F1 = FA and the
vertical line F2 = FB and has an overall intensity determined by the efficiency of transfer of
magnetization from nucleus A to nucleus B. This efficiency tells us something about the
relationship (J value or NOE intensity) between the two nuclei within the molecule.
Simultaneously with the process we described, other pairs of nuclei are undergoing the same
evolution, mixing and detection process resulting in other crosspeaks at the intersections of the
appropriate chemical shift lines and with characteristic intensities representing the efficiency of
transfer of magnetization. The 2D spectrum thus represents a complete map of all interactions
which lead to magnetization transfer, with the participants in the interaction addressed by their
chemical shifts.
Figure 1. 2D Matrix after Fourier transform in F2. The numbers represent data values at various cells in the data matrix, with the blank cells having zero or very small values.
All of the 2D experiments in use can be classified by the two types of nuclei detected in
the direct (F2) and indirect (F1) dimensions and the criteria for magnetization transfer during the
mixing step. The mixing pulse sequence is designed to select for certain types of interactions
between nuclei, and can be divided into two categories: magnetization transfer based on a Jcoupling
interaction, and magnetization transfer based on an NOE interaction.
Name F1 nucleus Mixing F2 nucleus
COSY 1H J 1H
RELAY 1H J, J 1H
TOCSY 1H J, J, J... 1H
NOESY 1H NOE 1H
HETCOR 1H 1JCH 13C
HMQC 13C 1JCH 1H
HMBC 13C 2,3JCH 1H
Experiments which transfer magnetization from one nucleus to another nucleus of the
same type, e.g., 1H to 1H, are called homonuclear experiments, and are characterized by a
diagonal defined by F1 = F2 and by pairs of crosspeaks at symmetrical positions across the
diagonal: F1=FA, F2=FB and F1=FB, F2=FA. This is because both magnetization transfer
pathways, 1HA --> 1HB and 1HB --> 1HA, can be observed. Experiments which transfer
magnetization
Figure 2. 2D matrix after Fourier transform in F1. The crosspeak at F2 = FB, F1 = FA is interpreted as an interaction between nucleus A and B which led to transfer of magnetization from nucleus A to nucleus B during the mixing period.
between two different types of nucleus, e.g., 1H and 13C, are called heteronuclear and lack a
diagonal and diagonal symmetry. The range of J values can be selected in mixing schemes: for
1H-13C couplings, 1JCH (the one-bond or direct coupling) is very large (125-160 Hz) while the
long-range 2,3JCH (two and three-bond coupling) is small (2-12 Hz). Mixing sequences can allow
multiple “jumps” of magnetization: RELAY is set up for two jumps based on J coupling - one
from the F1 nucleus to an intermediated (undetected) nucleus and a second jump to the F2
nucleus. The TOCSY experiment allows for many jumps, thus spreading the magnetization out
over an entire “spin system” or group of protons interconnected by J couplings. From these
basic experiments have grown many variants. The COSY has been extended to DQF-COSY
(reduced diagonal and improved phase properties) and COSY-35 (simplified crosspeak structure
for J-value determination). A common variant of NOESY is the ROESY, which gives better
results for molecules in the size range of peptides, oligosaccharides and large natural products.
HSQC gives the same results as HMQC but has better relaxation properties for large molecules
such as proteins. All 2D experiments and even 3D and higher-dimensional experiments are
based on the above list of basic 2D experiments.
In all this complexity of acronyms it is easy to forget that all 2D experiments do the same
thing: they allow you to correlate two atoms (nuclei) in a molecule based on an interaction
which is either through-bond (J-coupling) or through-space (NOE). The two nuclei are identified
by their chemical shifts, and the correlation appears in the 2D spectrum as a crosspeak at the F1
chemical shift (“y coordinate”) of the nucleus where magnetization starts and the F2 chemical
shift (“x coordinate”) of the nucleus to which the magnetization is transferred. Thus the basis of
all 2D experiments is the “jump” or transfer of magnetization. The information (J value or NOE
intensity) can be used to define structural relationships (dihedral angle or distance) but is only
useful if we can unambiguously assign the two chemical shifts (F1 and F2) to specific positions
within the molecule.
Bruker DRX Parameters for 2D NMR. Bruker generally uses the same parameter
names (sw, td, si, wdw, ssb, etc.) for both (F2 and F1) dimensions of a 2D experiment. In the
acquisition parameter menu (eda) and the processing parameter window (edp) there are two
columns of parameters, the left-hand column for F2 parameters and the right-hand column for F1
parameters. You can also specify parameters at the command line by entering the parameter
name (for the F2 dimension) or a 1 followed by the parameter name (for the F1 dimension). For
example:
sw 7 <return>
1 sw 150 <return>
would set the spectral width to 7 ppm in the F2 dimension and 150 ppm in the F1 dimension. The
offset parameters which set the center of the spectral window are exceptions to this rule – they
are named according to the RF channel used in each dimension. For example, the HSQC
experiment is a heteronuclear 2D experiment: the observe (F2) dimension is 1H and the indirect
(F1) dimension is 13C. Bruker designates these two RF channels as f1 (1H) and f2 (13C). This
refers to RF frequency 1 and RF frequency 2, not to be confused with the chemical shift scales F2
and F1 in the 2D spectrum. For this example the center of the spectral window in the F2 (1H)
dimension is set by the parameter o1 (offset for f1 channel in Hz) or o1p (in ppm). The center of
the spectral window in the F1 (13C) dimension is set by the parameter o2 (offset for f2 channel in
Hz) or o2p (in ppm). This is pretty confusing but at least it’s consistent: the f1 (first RF
channel) is always the observe (F2) dimension. In the case of homonuclear 2D experiments (e.g.,
DQF-COSY), there is only one RF channel: the proton (observe = f1) channel. In this case o1
(in Hz) and o1p (in ppm) determine the center of the spectral window in both the F2 and the F1
dimensions.
To understand 2D data processing, you will have to review the techniques of window
(multiplier) functions and zero-filling. These are 1D data processing techniques but they are
much more commonly used in 2D processing. Because of the large file sizes associated with 2D
experiments, it is sometimes necessary to use a small number of data points (td) in the acquired
FID. This can lead to the phenomenon of truncation, which means that the FID is cut off before
the signal has decayed to zero. Truncation is even more common in the t1 (indirect) dimension,
because each data point requires a complete acquisition of ns transients and time limitations
usually make it impossible to acquire the full t1 FID. One consequence of truncation is that the
transformed spectrum has too few data points: peaks and multiplets are poorly defined because
the digital resolution (number of data points per Hertz) is so low that only one or two data points
are available to describe the shape of a peak in the spectrum. The digital resolution of the
spectrum can be improved by a process called zero-filling. The FID is extended from td/2 data
pairs (complex pairs) to si ("size") data points by simply adding zeroes as data values for the new
data points. Fourier transformation of the new, larger FID dataset gives a spectrum with many
more points spread out evenly over the spectral width (Fig. 3). Note than the value of si is
limited to powers of 2 (e.g., 512, 1024, 2048, 4096, etc.).
Figure 3. The truncated FID is extended in time by adding new data points with zero intensity values. Fourier transformation yields a spectrum with the same number of data points as the extended FID. In order to avoid "wiggles" around each peak in the spectrum which result from the abrupt truncation of the FID, a window (multiplier) function is used to force the FID data to come smoothly to zero at the end of the acquired data.
One problem remains when a truncated FID is zero-filled and Fourier-transformed. The
abrupt drop at the end of the acquired data (td points) to the zero level of the added data points is
converted by Fourier transformation into a series of wiggles around the peak in the spectrum.
These "truncation wiggles" can extend outward and interfere with other peaks in the spectrum.
To eliminate the truncation wiggles, a window or multiplier function is applied to the acquired
FID data to force the FID to come smoothly to zero at the end of td data points. Two
commonlyused window functions which accomplish this are the sine-bell and the cosine-bell (or
90o shifted sine-bell) functions (Fig. 4). The cosine-bell function starts at the maximum (sine of
90o) at the beginning of the FID and goes smoothly to zero (sine of 180o) at the end of the
acquired data. This window function is commonly used for HETCOR, NOESY, and TOCSY
data. The sine-bell function starts at the zero point of the sine function (sine of 0o) at the start of
the FID data, reaches a maximum halfway through the FID (sine of 90o) and falls back smoothly
to zero by the end of the acquired FID data (sine of 180o). Because data later in the FID (in the
center) is emphasized over data early on in the FID, this window function leads to resolution
enhancement at the expense of sensitivity (signal-to-noise ratio). It is commonly used for DQFCOSY
data where resolution enhancement is necessary to prevent cancellation of nearby peaks.
Another commonly used window function is the sine-squared function, which is just the square
of the sine function. It rises more slowly at the beginning and drops off more gradually at the
end, giving more sensitivity enhancement (when shifted 90o) and better peak shape (suppression
of truncation wiggles) than the sine function.
Figure 4. The two most
common window functions for 2D NMR are the sinebell, which gives resolution enhancement at the expense of sensitivity, and the cosine-bell (or 90o shifted sine-bell), which gives sensitivity enhancement with some line-broadening. Both
functions come to zero smoothly at the end of the acquired FID data,
eliminating the truncation wiggles in the spectrum. The Bruker parameter
wdw determines the shape of the window function and ssb determines the shift.
Bruker uses the parameter wdw to describe the type of window function, for example
sine for a simple sine-bell function and qsine for a sine-squared function. A second parameter,
ssb, describes the amount by which the standard sine-bell (0o to 180o) is shifted at its starting
point. If ssb is set to zero, the sine-bell starts at the 0o point of the sine function. If ssb is set to
2, the window function starts at 90o (180 / 2) and ends at 180o. If ssb is set to 4, the window
function starts at 45o (180 / 4) of the sine function and ends at 180o. In all cases, the function
comes to zero smoothly, ending at the 180o point of the sine function.
The units used for FIDs and spectra can be very confusing. First and foremost, all of
these units are based on the data points by which the information is digitized. Since these points
generally come in complex pairs (real, imaginary), the number of data points may refer to the
total number of points (td) or to the number of complex pairs (td/2). For example, if td = 2048
and si 2048, the FID is zero-filled from 1024 (td/2) complex pairs to 2048 (si) complex pairs and
then Fourier-transformed to give a real spectrum (2048 data points) and an imaginary spectrum
(2048 data points). After phasing, the imaginary data points are discarded so that only the real
points are displayed, resulting in a spectrum display which contains 2048 (si) data points. The
FID can also be considered in terms of time units, or seconds. Each data point represents a time
increment corresponding to the dwell time (dw = 1 / (2*sw(F2)) in F2, in0 = 1 / (2*sw(F1)) in F1)
of the acquisition, and the entire FID stretches from time zero (start of the FID) to the acquisition
time (aq in F2, td(F1)*in0 in F1, end of the FID). After Fourier transformation, which converts
from time domain to frequency domain, the units of the spectrum are either Hertz or ppm (ppm =
Hz / spectrometer frequency). The width of the spectrum in ppm is always equal to the spectral
width, sw. The zero point of the ppm (or Hz) scale of a spectrum is set by setting the reference
frequency of some standard peak in the spectrum (e.g., TMS). It is important to remember,
however, that underlying all of the time and frequency units is the fundamental digitization of the
data, measured in data points or complex pairs of data points.
The 2D data matrix can be viewed in units of data points in both dimensions. Figure 5
shows the data matrix midway through data processing, with each t2 FID transformed and placed
into the matrix as an F2 spectrum. The number of FIDs actually acquired in the indirect (t1)
dimension is td in F1, leading to td(F1)/2 complex pairs. The actual vertical size of the matrix is
si data points which result after zero-filling from the td(F1)/2 complex pairs acquired in t1. Each
peak in the F2 spectrum results in a t1 FID extending vertically into the acquired data for td(F1)/2
data points, extended to a total size of si points by adding zeroes at the top of the matrix. Fourier
transformation of each column of the matrix yields the completed 2D data matrix, with
crosspeaks at the F1 frequencies which were represented in the t1 FIDs. The total number of data
points in the 2D data matrix is si(F2) * si(F1), and since each data point requires four Bytes, the
matrix occupies si(F2) * si(F1) * 4 Bytes of disk space. Because each Fourier transform produces
a real spectrum and an imaginary spectrum, there are actually four 2D matrices saved after the
double Fourier transform (xfb command). These are saved as files 2rr, 2ri, 2ir and 2ii (rr = real,
real; ri = real, imaginary; etc.). Phase correction in two dimensions is just a process of making a
linear combination of these four matrices using the phase parameters phc0 and phc1 in F2 and in
F1 to generate a “real, real” matrix which has absorptive peaks in both dimensions. This is a lot
of data: for a 2048 x 1024 point 2D matrix, the four matrices occupy a total of 2048 x 1024 x 4 x
4 = 33,554,432 Bytes. Because these datasets can fill up disk space very quickly, we have a
“daemon” which deletes processed 2D data which is more than 3 days old. To regenerate the 2D
data matrix, you only need to enter the xfb command
.
Figure 5. 2D Matrix represented in units of data points. Zero filling in t2 increases the FID size from td(F2)/2 to si(F2) complex pairs, which results in a spectrum of si(F2) data points after Fourier transformation and phase correction. Zero filling in t1 extends the FID from td(F1)/2 to si(F1) complex pairs, which results in an F1 spectrum of si(F1) data points after Fourier transformation and phase correction. The dimensions of the final 2D spectrum in
frequency units (Hz) are sw (F2) in F2 and sw(F1) in F1.
The following table summarizes the Bruker 2D parameters. In States mode (MC2 =
States) two FIDs (real and imaginary) are acquired for each t1 value, so that td(F1) represents the
number of complex pairs, or half of the number of FIDs acquired and saved. For TPPI mode
(MC2 = TPPI), one FID is acquired for each t1 value and td(F1) represents the number of FIDs
acquired and saved.
Bruker 2D Parameters
Direct (F2) Indirect (F1)
Acquisition Parameters:
Td td Number of data points to be acquired in the FID
Sw sw Spectral width in ppm
Dw in0 Time delay between successive time samples
aq in0*td(F1) Acquisition time in seconds
o1p o2p Center of spectral window in ppm (heteronuclear)
o1p o1p Center of spectral window in ppm (homonuclear)
nd0 Factor for calculating in0 (in0 = 1 / (nd0*sw(F1))
Processing Parameters:
si si Number of complex pairs after zero-filling
wdw wdw Window function (sine = sin, qsine = sin2)
ssb ssb Starting point of sine-bell function = 180o / ssb
phc0 phc0 Zero-order phase correction (chem.. shift independent)
phc1 phc1 First-order phase correction (chem.. shift dependent)
mc2 Phase-encoding method for F1 (States, TPPI, etc.)
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