Precision measurement of the quadrupole coupling and chemical shift tensors of the deuterons in α-calcium formate

Article abstract of DOI:10.1006/jmre.2001.2345

Using calcium formate, α-Ca(DCOO)₂, as a test sample, we explore how precisely deuteron quadrupole coupling (QC) and chemical shift (CS) tensors Q and σ can currently be measured. The error limits, ±0:09 kHz for the components of Q and ±0:06 ppm for those of σ, are at least three times lower than in any comparable previous experiment. The concept of a new receiver is described. A signal/noise ratio of 100 is realized in single-shot FT spectra. The measurement strategies and a detailed error analysis are presented. The precision of the measurement of Q is limited by the uncertainty of the rotation angles of the sample and that of σ by the uncertainty of the phase correction parameters needed in FT spectroscopy. With a 4-sigma confidence, it is demonstrated for the first time that the unique QC tensor direction of a deuteron attached to a carbon deviates from the bond direction; the deviation found is (1:2±0:3°). Evidence is provided for intermolecular QC contributions. In terms of Q, their size is roughly 4 kHz. The deuteron QC tensors in α-Ca(DCOO)₂ (two independent deuteron sites) are remarkable in three respects. For deuterons attached to sp² carbons, first, the asymmetry factors η and, second, the quadrupole coupling constants C_Q, are unusually small, η₁ = 0:018; η₂ = 0:011, and C_Q1 = (151:27±0:06) kHz, C_Q2 = (154:09±0:06) kHz. Third, the principal direction associated with the largest negative QC tensor component lies in and not, as usual, perpendicular to the molecular plane. A rationalization is provided for these observations. The CS tensors obtained are in quantitative agreement with the results of an earlier, less precise, line-narrowing multiple-pulse study of α-Ca(HCOO)₂. The assignment proposed in that work is confirmed. Finally we argue that a further 10-fold increase of the measurement precision of deuteron QC tensors, and a 2-fold increase of that of CS tensors, should be possible. We indicate the measures that need to be taken.

Full text of DOI:10.1006/jmre.2001.2345

Journal of Magnetic Resonance 151, 65–77 (2001)
doi:10.1006/jmre.2001.2345, available online at http://www.idealibrary.com on
Precision Measurement of the Quadrupole Coupling and Chemical
Shift Tensors of the Deuterons in �-Calcium Formate
Heike Schmitt, H. Zimmermann, O. K o¨ rner, M. Stumber, C. Meinel, and U. Haeberlen
Max-Planck-Institut f u¨ r Medizinische Forschung, AG Molek u¨ lkristalle, Jahnstrasse 29, 69120 Heidelberg, Germany
Received November 22, 2000; revised April 4, 2001
Using calcium formate, � -Ca(DCOO) , as a test sample, we ex- of molecular dynamics. Now, just before closing down our lab-
2
plore how precisely deuteron quadrupole coupling (QC) and chem- oratory, we wanted to find out how precisely we can measure
ical shift (CS) tensors Q and � can currently be measured. The
such tensors and what actually dominates the error limits. Per-
haps not surprisingly, it turned out during this study that the
increased precision allows us to address questions about which
people could so far at best speculate.
In the 1980s, several research groups (1–3 and others) started
to use extensively the quadrupole moment of the deuteron to
probe the structure and the dynamics of solid materials, often
materials of claimed biological interest. As a rule, these groups
error limits, � 0.09 kHz for the components of Q and � 0.06 ppm
for those of �, are at least three times lower than in any comparable
previous experiment. The concept of a new receiver is described. A
signal/noise ratio of 100 is realized in single-shot FT spectra. The
measurement strategies and a detailed error analysis are presented.
The precision of the measurement of Q is limited by the uncer-
tainty of the rotation angles of the sample and that of � by the
uncertainty of the phase correction parameters needed in FT spec-
troscopy. With a 4-sigma confidence, it is demonstrated for the first studied powder samples. Almost invariably, their analyses of
time that the unique QC tensor direction of a deuteron attached to 1d and 2d (exchange) spectra involved the tacit or outspoken
a carbon deviates from the bond direction; the deviation found is assumptionthattheQCtensorQatthesiteofadeuteronisaxially
�
(
1.2 � 0.3 ). Evidence is provided for intermolecular QC contribu-
symmetric about the deuteron’s bond and that the quadrupole
tions. In terms of Q, their size is roughly 4 kHz. The deuteron QC
tensors in � -Ca(DCOO) (two independent deuteron sites) are re-
markable in three respects. For deuterons attached to sp carbons,
first, the asymmetry factors � and, second, the quadrupole coupling
Q 1 2
constants C , are unusually small, � = 0.018, � = 0.011, and
2
coupling constant CQ = e qQ/h (we use Slichter’s notation
2
(4)) is, depending on the type of bond, some value between
2
1
44 and 193 kHz (5). Until very recently, such assumptions
were even made in cases (6) where the complete QC tensor Q
had been measured before in single crystal samples (7). For
spin-1 nuclei such as deuterons it is convenient to define the
C
Q1 = (151.27� 0.06) kHz, CQ2 = (154.09� 0.06) kHz. Third, the
principal direction associated with the largest negative QC tensor
3
QC tensor by Q = 2 (eQ/h)V where h is Planck’s constant
component lies in and not, as usual, perpendicular to the molecular
plane. A rationalization is provided for these observations. The CS and e the (positive) elementary charge. Q is the (scalar) nuclear
�
30
2
tensors obtained are in quantitative agreement with the results of quadrupole moment which for deuterons is +0.2860 � 10
m
an earlier, less precise, line-narrowing multiple-pulse study of � - (8). Vistheelectricfieldgradient(EFG)atthesiteofthenucleus.
2
Ca(HCOO) . The assignment proposed in that work is confirmed.
This definition ensures that the separation of the components
of the quadrupolar split deuteron doublet in a high-field NMR
experiment is directly given by Qzz where the z-direction is
parallel to the applied magnetic field B0.
The justification of the assumptions about the shape, orien-
tation, and size of a deuteron’s QC tensor comes from three
sources. The first are quantum chemical calculations which pre-
dict, for deuterons bound, e.g., to carbons, more or less axi-
ally symmetric QC tensors aligned along the deuteron’s bond
Finally we argue that a further 10-fold increase of the measurement
precision of deuteron QC tensors, and a 2-fold increase of that of
CS tensors, should be possible. We indicate the measures that need
to be taken. �C 2001 Academic Press
Key Words: precision deuteron quadrupole coupling/chemical
shift tensors; relation to n-structure.
INTRODUCTION
(9). With regard to the size of CQ, the predictive power of
Wearewellawarethataprecisionmeasurementofquadrupole such calculations is modest, in view of the narrow range in
coupling (QC) and chemical shift (CS) tensors in a compound which CQ is found in different compounds and in different
as simple and nonbiological as deuterated �-calcium formate, types of bonds it is, in fact, almost worthless. The second
Ca(DCOO)2, is against the solid-state NMR mainstream in source is just the wealth of experience that has accumulated
the dawning 21st century A.D. For almost 3 decades we have over the course of time. There is no doubt: the assumptions
measured proton and deuteron CS tensors and, for 2 decades, work. The third source is a limited number of single crystal
deuteron QC tensors, often as a necessary first step in studies measurements of deuteron QC tensors done mostly, but not
65
1
090-7807/01 $35.00
Copyright �C 2001 by Academic Press
All rights of reproduction in any form reserved.

6
6
SCHMITT ET AL.
exclusively, in our laboratory, and comparisons of the orientation as we will see, the final errors of Q are not dominated by our
of Q with structure information derived from X-ray and, in par- limited ability to locate the centers of gravity of the resonances.
ticular, neutron diffraction experiments (10–17). Generalizing, The sharpness of the resonances allows us as well to find the cen-
with due caution, the results of these studies, the above-quoted ters of gravity of the quadrupolar split doublets with sufficient
assumptions can be quantified in the following empirical rules accuracy for a measurement of the deuteron CS tensors. The
(i)–(iii): First, the QC tensors of deuterons bound to carbons, precision achieved exceeds by three times the precision of a for-
oxygens, . . . , are nearly axially symmetric. Thus, rule (i) states: mer proton line-narrowing multiple-pulse NMR experiment on
the asymmetry factor � (4) of such tensors remains in the range �-Ca(HCOO) (23, 24) which, in terms of the error limits of the
2
0
� � < 0.1. The upper end of this range tends to be approached obtained CS tensors, is actually the very best such experiment
�
whenever the deuteron is part of a pronounced planar structure ever performed.
(16, 17).
The space group of �-calcium formate is orthorhombic (Pcab)
Second, because of the near-axial symmetry of Q, this tensor with Z = 8 (25), meaning that the spectral information obtain-
has what we call a unique principal direction, denoted by eZ . It able from rotating only one sample crystal about an axis perpen-
is the principal direction of Q which is parallel to the approx- dicular to the applied field B is not only sufficient but already re-
0
imate axial symmetry axis and corresponds, consequently, to dundant for the measurement of the QC and CS tensors. We will
the largest principal component, QZ Z , of Q (by magnitude, but exploit this redundancy to determine within a few hundredths of
actually also in absolute terms (13, 18)). With regard to struc- a degree the direction of the rotation axis of our sample crystal.
tural and dynamical applications of deuteron NMR, the crucial A careful structure determination of �-Ca(HCOO) by neutron
2
proposition is that the unique principal direction eZ of the QC diffraction at T = 100 K and T = 296 K has been performed
�
tensor is, within less than 1 , parallel to the deuteron’s bond in 1977 by Burger et al. (26) and another by Bargouth and Will
direction, eb. This is rule (ii). Third, if the deuteron is part of a in 1981 (27). According to the error limits specified in (26) and
planar structure, e.g., of an aromatic ring, the asymmetry factor (27), the positional parameters in (27) are less accurate by a fac-
�
of Q will substantially deviate from zero. It is then meaningful tor of 2–3 than those in (26). In what follows, we will therefore
to consider, apart from eZ , also the two other principal directions refer ourselves to the data of Burger et al. Figure 1 shows views
eX and eY . Adopting the convention |QZ Z | > |QX X | � |QYY |, of the unit cell of �-Ca(HCOO) looking down the c and a axes.
2
rule (iii) states that eX is perpendicular to the plane in question. The asymmetric unit holds two independent formate anions; �-
The accuracy of this rule is not as good as that of rule (ii). It calcium formate thus offers the chance to test rules (i)–(iii) at
�
appears to be valid within, say, 10 (16).
two distinct deuteron sites.
Note that both rules (ii) and (iii) imply that, compared to the
We will find, for the first time for a carbon-bound deuteron,
QCcontributionsoftheimmediatebond, theintermolecularcon- a deviation of e from e outside the combined error limits of
Z
b
tributions are small, but it is hard to say how small they are really. the NMR and neutron diffraction experiments. We will also find
To verify or falsify rule (ii) for a given compound calls for that, for both formate ions, rule (iii) is not valid. It turns out
measuring eZ and, independently, eb with a precision of better that e , and not e , is the principal direction of Q which is
Y
X
�
than 1 . With single crystal NMR it is no serious problem to perpendicular to the molecular plane. We will provide a natural
meet this requirement for eZ . On the other hand, our experience reason for this result and, in so doing, also a natural reason for
with X-ray data for eb is that the error limits are hardly less the usual validity of rule (iii).
�
than � 5 even in cases where the published limits are much
smaller. Examples of obvious discrepancies of this kind can be
EXPERIMENTAL
found in Refs. (12, 19) and (15, 20). Hence, the only method
for getting deuteron bond directions with sufficient accuracy to
test rule (ii) meaningfully is neutron diffraction. Still, we are
Sample Preparation
not aware of any case where a nonzero deviation of eb and eZ
2
The synthesis of Ca(DCOO) was started by preparing deuter-
outside the combined error limits of the neutron diffraction and ated formic acid. This was done by repeated exchange of oxalic
NMR experiments has been found (note, we are not talking of acid dihydrate, (COOH) � 2H O, with D O and drying the
2
2
2
deuterons in hydrogen bonds where such deviations have at least product, deuterated oxalic acid, under reduced pressure to get
been claimed (21, 22)).
deuterated anhydrous oxalic acid, (COOD) , followed by ther-
2
�
Calcium formate is an excellent candidate for testing rules mal decomposition at 160–180 C to get perdeuterated formic
(
i)–(iii). From an aquous solution beautiful crystals of the �- acid, DCOOD. Neutralization with CaCO and evaporation to
3
form can be grown which show no tendency to weathering even dryness yielded Ca(DCOO) . Crystals were grown from a sat-
2
�
after years in normal atmosphere. The unusual dilution of the urated solution in H O at 40 C (the carbon-bound deuteron
2
deuterons in �-Ca(DCOO)2 and the virtual absence of any other of the formate ion does not exchange!) and slow evaporation
magnetic nuclei lead to exceptionally sharp lines in the deuteron at the same temperature under constant stirring. Diamond-like
NMR spectra (see below). This allows us to determine the crystals of �-Ca(DCOO) with diameters of up to 7 mm were
2
deuteron QC tensors Q with unprecedented precision. Actually, obtained after 2 months. Using an optical goniometer, some of

2
PRECISION MEASUREMENT OF H QC AND CS TENSORS
67
tal was machined on a lathe to a diameter of 4.2 mm, fitting
precisely into a standard 5-mm NMR tube cut to an appropriate
length (44 mm) for the NMR probe. The rod with the crystal
was fixed inside the NMR tube with a droplet of epoxy glue.
Apparatus
With one modification we used the same FT spectrometer that
served us in many deuteron NMR studies during the past decade,
e.g. (7, 28). It is built around a narrow-bore 11-T Bruker cryo-
�
9
magnet whose driftrate fell over the years well below the 10 /h
level, which is important for the present project. The modifi-
cation concerns the receiver. The new receiver (29)—christened
ADC2000 by our former graduate students because it took a long
time to complete it and they knew U.H. would be forced to retire
in late 1999—features a local oscillator frequency �loc for the
phase-sensitive detector (mixer) which is lower than the trans-
mitter frequency �tr (� �L = 72.12 MHz) by �tr/36 � 2 MHz.
This means that an on-resonance deuteron NMR signal appears
at the output of the receiver as a (decaying) 2-MHz signal. To
prevent noise from the mirror frequency at about 68 MHz to
appear, after the mixer, in the 2-MHz band, a steep three-section
7
2
2-MHz filter with a flat top of 2-MHz and an attenuation of
6 db at 68 MHz is placed between the preamplifier and the
receiver. The 2-MHz signal after the mixer is amplified by an
ac amplifier with a virtually flat gain of 20 db from 1 to 3 MHz,
dropping to 0 db at 5 MHz and cutting off still higher frequen-
cies. The 2-MHz NMR signal is sampled by a 12-bit, 10-MHz
(actually �tr � 5/36 � 10.017 MHz) analog-to-digital converter.
In steps of two, up to 128 K samples can be taken corresponding
to a signal duration of 13.1 ms. This was judged sufficient for
solid-state deuteron NMR, and as it turned out, it is sufficient.
Signal accumulation requires that the initial phase of all FIDs to
be accumulated is the same. To ensure this requirement, the rise
of the RF pulse which excites the FID (or the rise of the pulses
whichexciteanecho)issynchronizedtothe�tr/36 � 2MHzsig-
nal. Accumulation is under the control of a TMS320C40 signal
processor. After completion of the accumulation, this processor
performs all necessary steps (including a real Fourier transfor-
mation) to generate a phase-corrected spectrum and transfers
the interesting part of the spectrum (that centered at 2 MHz and
covering a selectable range of, e.g., � 156.25 MHz) to the host
computer for storage on a disc and to a display for visual in-
spection. The entire software of the signal processor finds room
in its cache memory. Therefore, despite the possibly rather long
data files (128 K), all computions are performed in a time which
is hardly noticeable by the user.
FIG. 1. View of the unit cell of �-Ca(HCOO)2 along a and along c following
26). One of the eight formula units is marked. Pairs of these units are related
by inversion centers and are thus magnetically equivalent. Hydrogens of type 1
and type 2 are shaded differently.
(
the larger of the harvested crystals were examined with regard
to the quality of the natural growth planes. A particularly beau-
tiful specimen was selected for NMR. Its natural growth planes
were indexed by measuring the angles �pq between the nor-
mals of all pairs of planes p and q and comparing these angles
(
n)
with the angles � between pairs of low-indexed reciprocal lat-
rs
�
�
tice vectors r and s calculated from the neutron diffraction
structure data (26). The crystal was fixed with two-component
epoxy glue (UHU plus endfest 300, made by UHU GmbH, B u¨ hl,
Germany) to a rod made of PVC such that the rod axis (which
subsequently was to become the rotation axis of the crystal in the
�
�
NMR goniometer) had polar angles � � 75 and � � 15 in the
ˆ
crystal-fixed axis system aˆ = a/a, b = b/b, cˆ = c/c where a,
The advantages of the ADC2000 over the former, traditional
receiver with quadrature phase detection are as follows:
b, and c are the primitive lattice vectors of the orthorhombic cell
˚
˚
˚
and a = 10.168 A, b = 13.407 A, and c = 6.278 A (26).
The orthorhombic symmetry of the crystal generates effec-
� The user does not need to care about the amplitude and
tively three more rotation axes such that rotation of the crystal phase balances of two orthogonal-phase channels. Any phase
about the specified axis provides access to much more than the cycling for counteracting imbalances becomes unnecessary.
minimum information necessary for measuring the QC and the
� Long-term drifts are excluded because only ac amplifiers
(symmetric constituent of the) CS tensor. The rod with the crys- are used.

6
8
SCHMITT ET AL.
�
The receiver contributes virtually nothing to the deadtime g—were machined on a milling machine which guarantees or-
of the spectrometer because no narrow video filters are involved thogonality of probe and goniometer axes within a fraction of a
that prevent, in traditional designs, “mirror” noise, potentially minute of arc. The tolerances of the bearings are such that they
�
generated in slowly running ADCs, from being added to the in- keep a possible wobbling of the NMR tube below � 0.03 . The
band noise. In the ADC2000 no compromises between deadtime worm of the gear is made of aluminum and the toothed wheel
and filters are necessary.
(36 teeth) of Vespel. Both pieces were made many years ago
�
The ADC2000 is tailored to the spurious-signal- by a firm which specialized in making gears. We expected that
suppression scheme described by Speier et al. (28). It is able their experts would produce a virtually perfect, evenly divided
to use different file lengths for the signal proper (as mentioned, toothed wheel. The NMR results reported below suggest that
up to 128 K) and for the spurious signal (e.g., 512 samples), our expectation was wrong. The worm is connected with a rod
allowing us to reduce the delay between the NMR-saturating, of aluminum to a knob and a dial with 100 divisions at the top of
spurious-signal exciting pulses to 2.5 ms. This means that the cryostat. When recording spectra for a rotation pattern, we
Speier’s scheme can be applied to samples with T1 as short as always rotated the knob/worm by one full turn corresponding
�
1
0 ms. Using this scheme, an undistorted signal can be recorded to a 10 rotation of the sample, and the dial allowed us to set
�
not later than 5 �s after the center (which marks the “true” time this angle to within � 0.01 (� 1/10 division of the dial). Inside
�
zero for the FID) of a 2-�s 90 pulse. The remaining dead time the toothed wheel the NMR tube is held by a rubber O-ring
is dominated by the bandwidth of the NMR probe.
which is pressed onto the tube by a hollow screw to prevent any
slip between the tube and the wheel. A 100-� platinum resis-
tor in the vicinity of the sample served to monitor the temp-
We should also mention some (minor) disadvantages:
�
Because of the synchronization requirements, the program- erature.
ming of the pulse generator (30) becomes somewhat more in-
volved, e.g., the spacing of pulses that affect the phase of the
NMR signal must be an integer multiple of 500 ns. Other times
such as the widths of pulses can remain on the usual 100-ns
raster of the pulse generator.
DATA PROCESSING AND RESULTS
Preparatory Tests
�
An on-resonance NMR signal is available as a 2-MHz sig-
A precision measurement of QC and CS tensors calls for
nal and not as a (decaying) dc signal. This is somewhat incon- recording NMR spectra for a fairly large number of crystal ori-
venient when adjusting widths of 90 or 180 pulses.
entations. In order to find out what the appropriate delay tw
Because the sampling rate of the ADC is tied to the spec- between RF shots is, we first measured the spin–lattice relax-
�
�
�
trometer frequency �tr, the frequency scale of the FT spectra ation time T1. We used the saturation-recovery method and chose
depends slightly on the setting of �tr. In the data analysis below, more or less arbitrarily the goniometer rotation angle for which
�
this dependence is of course taken into account.
B0 lies about 5 off the ab plane of the crystal. From a series
of spectra recorded for a range of delays tw, we concluded that
A similar receiver concept is realized in the Avance spec- a certain doublet (actually the innermost) is the slowest to re-
trometers of the Bruker company. In those spectrometers, the cover, but the others are only marginally faster. T1 of this doublet
sampling rate is, however, substantially smaller than in our was measured and found to be (37 � 3) min at the temperature
ADC2000, and the delicate synchronization problems discussed T = (298 � 0.5) K. The size of T1 suggests that it is spin diffu-
abovearesolvedinadifferentway. Also, becauseofthelowsam- sion (32) which is responsible for the almost uniform relaxation
pling rate, quadrature phase detection is still required to avoid of all distinct deuteron spins in Ca formate. The orthorhombic
mirror noise (31).
crystal structure in combination with spin diffusion also suggests
The spectrometer component that is crucial for the precision that T1 is only weakly dependent upon the crystal orientation.
of the measurement of QC and CS tensors is the NMR probe As a result of this test we decided to choose, for the final tensor
and, in particular, its goniometer.
measurements, tw not shorter than 1 h.
The idea is that the axis g of the goniometer is perpendicular
Next we tested how many accumulations are needed to obtain
to the applied field B0. However, we do not know the direction spectra with an acceptable signal-to-noise (S/N) ratio. In Fig. 2
of B0. No NMR spectroscopist does. What is available as a ref- we show two spectra. The upper is the Fourier transform of a
erence is the innermost tube of the magnet’s cryostat and we can single FID with the spin system having been in thermal equilib-
only assume that this tube is parallel to B0. The cylindrical NMR rium at 298 K prior to the 2-�s RF excitation pulse. The lower
probe is 180 mm long and 50 mm in diameter. We wrapped tape is an overnight spectrum from 10 accumulated FIDs. In passing,
around its ends such that the probe fits smoothly but tightly into we note that the quality of these spectra should give credit to the
the cryostat. The drive of the goniometer consists of a worm performance of the ADC2000. In the single shot spectrum the
gear. The bearing of the toothed wheel (which holds the NMR average width �� of the lines is 667 Hz, substantially narrower
tube with the sample) and that of the NMR tube at the opposite than for “typical” deuterated compounds, and the S/N ranges
side of the probe—these bearings define the goniometer axis from 83 to 116, the average being 100. We shall see below that it

2
PRECISION MEASUREMENT OF H QC AND CS TENSORS
69
result of this test was that � increased from � 2.5 Hz (perfect
�
phase) to � 7.1 Hz (1 phase error) with the sign of � being
correlated, of course, with that of the phase error. Below we
will see that the latter number for � is realistic. It corresponds to
only one-hundredth of the linewidth and to only one-twentieth
of the digital resolution ��dig. Essentially the same procedure
for locating resonances as used here was adopted in (22). The
authors estimate that their precision is about 50 Hz. Looking
at the spectra in their Fig. 1 reveals that the phase correction
problem was not solved satisfactorily.
The heights of the lines in Fig. 2 differ somewhat from each
other. This is caused, on the one hand, by differences of the
linewidths �� and, on the other hand, by the digital resolution
which is 152.8484 Hz. Note that the lines of the outermost dou-
blet, whose splitting is 228.4 kHz, are among the highest. This
demonstrates the adequacy of both the excitation and the receiv-
FIG. 2. Deuteron NMR FT spectra of a single crystal of �-Ca(DCOO)2.
Top: single-shot spectrum; bottom: overnight spectrum, 10 FIDs accumulated.
The rotation angle � refers to Fig. 4. A manual, two-parameter phase correction ing bandwidths of the spectrometer.
�
with a resolution of 1 was applied. From the scatter of the CS data points in
Fig. 5 it follows that the centers of gravity of the resonances in these spectra can
be located within � 7 Hz. The digital resolution is 152.8484 Hz. The spurious
out-of-phase signals on the left end of the single shot spectrum are from an
unknown extraneous source. Due to cancellation, they are virtually absent in the
overnight spectrum.
Finally, in order to test the mechanical precision of the go-
niometer and its drive, we compare in Fig. 3 spectra recorded
for the rotation angles � = 0 , 180 , and 360 . If the goniome-
ter, the drive, and the orthogonality of g and B0 were perfect,
the quadrupole splittings 1�Q in all of these spectra had to be
�
�
�
�
�
�
identical. In the 0 and 360 spectra they indeed agree satisfac-
torily; in the 180 spectrum, however, they deviate deceivingly
�
is large enough as not to be a limitation to the precision of the QC
tensor measurements. By fitting gaussians to the lines we locate
their centers of gravity. Theoretically, the standard deviation �
associated with this procedure is (33)
much from those in the other two spectra. Actually, for � = 0 ,
B0 lies close to the bc plane of the crystal and the usually eight
distinct doublets become pairwise degenerate. An inspection of
the rotation pattern in Fig. 4 (to be discussed below) reveals that
doublets 1 and 3 in Fig. 3 (we count them starting from the in-
�
= � const ��/(S/N).
[1] nermost doublet) are sensitive to a (small) misorientation, while
�
doublets 2 and 4 are insensitive because, for � = 0 , the path
Extensive numerical tests (14) yielded const = 0.34 � 0.06. of B0(�) either crosses the equator (doublet 2) or passes near
We wondered whether the digital resolution ��dig should also the pole (doublet 4) of the respective QC tensor. The splitting
show up in [1] and therefore repeated the numerical tests with 1�Q of doublet 1 depends most strongly on �. For this doublet,
d1�Q
d�
�
the same ratio of ��/��dig as in the actual experiment. These
tests confirmed that const � 0.34. By inserting this value in [1]
together with S/N = 100 and �� = 667 Hz, we get the amaz-
ingly small number of � 2.3 Hz for �.
|�=0�,180� = 4.16 kHz/ . The measured splittings of this
In FT spectroscopy, however, there is, in addition to noise,
another unavoidable source of error in locating the center of
gravity of a resonance line. This is the phase-correction step
in the data analysis. For that step, we used Heuer’s automatic
phase-correcting routine (34). Visual inspection of the spectra
so obtained conveyed the impression that the phase correction
parameters cp and lp (see (34) for their definition) proposed by
the routine can still be improved. Therefore, we refined these
parameters manually. The quality of the spectra allowed us to
�
find the best values for cp and lp within 1 , and this was also the
digital resolution of the manual phase control. In order to learn
what the consequence of an erroneous phase correction is, we
generated synthetic noisy time data (S/N = 100, �� � 670 Hz,
�
�/��dig � 4.5, all values as in the experiment), introduced, after
�
�
�
FIG. 3. Test of goniometer settability (� = 0 , 360 ) and of 180 -
�
Fourier transformation, a phase error of � 1 and determined
�
periodicity (� = 180 ) of the spectra. See text for details and quantitative as-
with the fitting routine the centers of gravity of the lines. The pects.

7
0
SCHMITT ET AL.
There is a second QC tensor contributing to doublet 1. Its
unique axis is strongly oblique to the goniometer axis g. The
splitting of the respective doublet is therefore sensitive to the
orthogonality of g and B . The fact that doublet 1 is as sharp
0
�
�
in the 180 spectrum as it is in the 0 spectrum indicates that a
possible nonorthogonality of g and B0 is not a serious problem.
What to do with the imperfect goniometer? The best solution
would obviously be to get a new gear and, after having verified
that the division of the toothed wheel is uniform, to repeat taking
data. Alas, it is too late for that. In Section IV of (35) we pointed
out that the problem with the unknown angle ^(g, B0) can be
alleviated by recording rotation patterns of line positions (which
are subsequently used for fitting QC and/or CS tensors) over a
full turn 0 � � � 2� and not, as usual, only over the “sufficient”
1
range 0 � � � �. By taking averages h1�Q(�)i := [1�Q(�)+
2
1
�Q(� +�)] and proceeding otherwise as usual, the remaining
FIG. 4. Rotation pattern of line positions. For � = �1, �2 and �3, the
applied field is in the ab, ac, and bc plane, respectively. Note the lack of perfect errors will be kept quadratically small in � := ^(g, B0) � �/2.
�
1
80 periodicity of the pattern. The full curves are fitted harmonic functions (not
We will now argue that exactly the same procedure allows us
to cope with a nonuniformly divided goniometer toothed wheel.
Denoting the nominal rotation angle by � and the true rotation
fitted quadrupole splittings). The scatter of the data points relative to the fitted
quadrupole splittings plus CSs corresponds to not more than a sixth (!) of the
width of the full curves.
˜
angle of the wheel by �, the nonuniformity can adequately be
modeled by
�
�
doublet in the 0 and 360 spectra are 58.49 and 58.57 kHz,
�
respectively. That means, after a 360 rotation, we reproduced
the initial crystal orientation within 0.019 which gives credit to
˜
�
(�) = � + ϕˆ sin(� � �c),
[2]
�
the settability of the drive of the goniometer.
On the other hand, the splitting of doublet 1 in the 180 spec-
where ϕˆ and �c are constants whose numerical values we may
find by examining pairs of spectra for general values of � and
�
trum is 61.95 kHz and thus 3.42 kHz larger than, on the average,
�
�
�
�
+ 180 , and not only those for � = 0 and � = 180 as we
�
�
in the 0 and 360 spectra. This implies that either the division
of the toothed wheel or the orthogonality of g and B0, or both,
is anything but perfect. As it turns out, the unique axis of one of
the two QC tensors contributing to doublet 1 is almost perpen-
dicular to g, meaning that the path of B0(�) leads from the QC
tensor equator right to its pole. This implies that all along that
path the splitting 1�Q is insensitive to a (small) excursion of B0
in the perpendicular direction, that is, to a (small) deviation of
�
�
did so far. In this way we got ϕˆ = 0.01 =b 0.6 and �c = 60 .
For any nominal angle � we might now calculate a correction
˜
1
� = �(�) � � and take this correction into account in the
tensor fitting procedure. Alternatively, we may calculate from
the raw data 1�Q (�) corrected quadrupole splittings
raw
d1�Q(�)
corr
raw
1
�
(�) = 1�Q (�) �
1�,
[3]
Q
�
d�
the angle ^(g, B0) from 90 . Therefore, we must conclude that
�
�
the major part of the differences in the � = 0 and � = 180
where, because 1�Q(�) = A + B cos 2(� � �0) with A, B, and
0 being constants,
spectra in Fig. 3 results from an error in the rotation angle �.
�
From the above-quoted numbers it follows that instead of the
�
nominal � = 180 rotation, the true rotation angle was only
�
�
�
(
180 � 0.82 ). The splittings of the other doublets in the 180
d1�Q(�)/d� = � 2B sin 2(� � �0).
[4]
spectrum and the beginning lifting of the degeneracy of the two
tensors contributing to doublet 3 support this conclusion, an- Equations [2]–[4] imply that 1� Qc orr(�) � 1� (�) is equal in
raw
Q
�
noying as it is. In the meantime we checked the uniformity of magnitude and opposite in sign for � and � + 180 . Therefore,
the division of a toothed wheel stemming from the same lot as within the validity of [2] and the linear approximation [3], the
corr
Q
raw
Q
that in our former deuteron NMR probe (remember, our lab was averages h1� (�)i and h1� (�)i are equal. Hence, by tak-
closed down in late 1999 and all equipment is gone). Using a ing such averages we may forget about the difference of raw and
light pointer and a two-sided mirror fixed to an NMR tube we corrected quadrupole splittings, or nominal and true rotation an-
�
could confirm that 360 rotations are reproducible within less gles. In this way all errors are eliminated which are linear not
�
�
than 0.1 . On the other hand, 180 rotations were consistently only in �, but also in ϕˆ . Below, we will adopt this procedure.
�
2
2
in error by roughly 0.3 . While this toothed wheel is obviously The remaining errors are of order ϕˆ and � . For ϕˆ = 0.01,
better than the specimen in the NMR probe, the nonuniformity the worst case error will be, in terms of the quadrupole split-
2
of its division is nevertheless much larger than anticipated.
ting, 2 ϕˆ CQ � 33 Hz. This (systematic) error is larger than the

2
PRECISION MEASUREMENT OF H QC AND CS TENSORS
71
(
d)
standard deviation of locating line positions, but smaller than the
error stemming from the uncertainty of setting the goniometer
dial. Nevertheless, it contributes significantly to the final mea-
surement error of the QC tensors. Due to the smallness of the
deuterons’ CS anisotropies, the imperfection of the goniometer
has no consequences for the precision of the measurement of the
CS tensors.
gravity � (� ) of all doublets. As suggested by the discussion
cg
n
above, we then took the averages
�
�
1�
�
(
Q
d)
(d)
Q
(d)
Q
�
1
� (�n) := 1� (�n) + 1� (�n + 180 )
2
and
�
�
1�
�
(
d)
(d)
cg
(d)
�
�
(�n) := � (�n) + � (�n + 180 ) .
Rotation Pattern of Line Positions and Data Analysis;
Quadrupole Coupling Tensors
cg
cg
2
(
d)
Q
Next, the red set of averages h1� (�n)i was subjected to
From our sample crystal of �-Ca(DCOO)2 we recorded
�
the tensor fitting program SUPERFIT. The original (Fortran)
version of this program was brought in 1979 to the MPI in
Heidelberg by J. Tegenfeldt from Uppsala. Since then it was
deuteron spectra for �n = n � 10 , n = 0, . . . , 36. During
daytime we chose tw = 1 h and recorded single shot spec-
tra. Overnight, 10 FIDs were accumulated. The sample tem-
perature was not regulated but monitored and found to be
T = (287 � 0.5) K during all the time of data taking. After
baseline (36) and phase correcting the spectra, the centers of
gravity of all resonances were located as described above and
written in a file. From this file, the rotation pattern shown in
Fig. 4 was generated. In order to find out which resonances arise
from the same nuclei in subsequent spectra, we fitted sin-curves
to the data points. These sin-curves are also shown in Fig. 4. In
+
+
version.1
updated several times and is now used in a C
One of us (U. H.) discussed it at NMR schools in Waterloo
1991) and Portoroz (1993) and even prepared lecture notes.
(
SUPERFIT is much more versatile and flexible than the still
often advertised Volkoff method (37) which stipulates that
quadrupole splitting or chemical shift rotation patterns from
three mutually orthogonal rotation axes have been recorded.
The ability of SUPERFIT to accept data from an arbitrary
set of orientations � ( := (ϑ , ϕ ) of B is essential to
B)
(B)
(B)
�
�
�
the range 0 � � � 180 there are three angles �1 = 48.57 ,
0
�
�
this work. SUPERFIT provides a least-squares fit of n =
�
2 = 134.41 , and �3 = 179.67 (the digits behind the decimal
(
B)
1
, 2, . . . , N � 5 measured quadrupole splittings 1�Q,n(�
)
n
point are obtained by a fitting procedure, see below) where all the
sin-curves cross pairwise. At these angles the path B0(�) crosses
the ab, ac, and bc planes, respectively, of the orthorhombic crys-
tal. By looking at which pairs of sin-curves cross at �1, �2, and
(
n
B)
or of n = 1, 2, . . . , N � 6 chemical shifts (�n(� ) � �L)/�L
to a traceless QC tensor Q, or a general symmetric second-rank
tensor �. In the fitting procedure, the orientations � ( of B0
B)
n
are assumed to be known exactly—which, of course, is not the
case. The tensors are specified by their cartesian components
Q�� and ��� in a crystal-fixed, but otherwise arbitrary orthogo-
�3, we can partition these curves in two sets which we label, for
convenience, as red and blue sets. Eventually, these sets must
be assigned to the two types of distinct deuterons in the crystal;
see below.
nal axes system 6 to which ϑ( and ϕ are also referenced.
For �-Ca(DCOO)2 it is natural to choose the axes of 6 along
a, b, and c. Because the measured quantities 1�Q,n and �n are
linear functions of the Q�� and ��� , respectively, least-squares
fitting is a noniterative process. Note that least-squares fitting
of second-rank tensors becomes a cumbersome iterative pro-
cess with the need of guessing starting values if the tensor is
specified by its principal values and the Euler angles relating
its principal directions to the axes of the reference frame, see,
e.g., (38).
B)
(B)
Thecrossingpoints�1, �2, and�3 playanadditionalroleinthe
data analysis: Above we stated that because of the orthorhombic
symmetry of �-Ca(DCOO)2 one rotation pattern with doublets
from four symmetry related deuterons is worth four rotation
patterns for one deuteron. The four rotation axes are generated
�
�
from the original one with polar angles ϑR � 75 , ϕR � 15 by
twofold rotations about the primitive axes of the crystal which
are all parallel to 21 axes. We are free to assign any one pair
of sin-curves (corresponding to some doublet in the spectrum)
from, say, the red set, to the original rotation axis. After that,
there is no freedom left. By looking at which individual sin-
curves cross at �1, �2, and �3, respectively, we may find out to
which of the rotated rotation axes the other sin-curves must be
assigned. For example, the sin-curve which crosses that assigned
to the original rotation axis at �1, i.e., where B0(�) is in the ab
plane, must be assigned to the rotation axis generated from the
The following feature of SUPERFIT assists in handling data
from rotation patterns and is useful for finding refined orienta-
tions of B0: Instead of specifying � ( for each measurement
B)
n
n, the direction ϑR, ϕR of the crystal’s rotation axis, a reference
direction ϑref, ϕref, and a reference angle �ref are given together
with the rotation angle �n. For � = �ref, B0(�) is parallel to
the projection of the reference direction onto the plane perpen-
�
dicular to the rotation axis. From ϑ , ϕ , ϑ , ϕ , � , and �
original one by a 180 rotation about the c axis.
R
R
ref
ref
ref
n
SUPERFIT calculates ϑ( and ϕn . For all data of a rotation
B)
(B)
After having decided which of the data points in Fig. 4 belong
to the red and which to the blue set, and which sin-curves go
with which rotation axis, we evaluated the splittings 1� (�n)
n
(
Q
d)
1
The source code of SUPERFIT is available on request. Contact Heike
(d = 1, . . . , 8 labels the doublets) as well as the centers of Schmitt; e-mail: heike@mpimf-heidelberg.mpg.de.

7
2
SCHMITT ET AL.
pattern, the angles ϑR, . . . , �ref are fixed parameters and only �n four symmetry related deuterons of type 1 (of type 2) of the
varies. In this work we chose as the reference direction the cross- crystal. This labeling of the deuterons follows (26). For conve-
ing of the path B0(�) with the ac plane. This choice links the nience, we list the tensors both in their cartesian form and by
reference direction to the direction of the rotation axis, namely their principal values and principal directions. The QC tensors
�
1
�
ϑref = tan (1/ tan ϑR cos ϕR) and ϕref = 180 and says that of the other deuterons in �-Ca(DCOO)2 are obtained from those
�
ref equals �2. Note that the specification of the reference direc- in Table 1 by inverting in the cartesian tensor forms the sign of
tion does not introduce any extra uncertainty. On the other hand, any two of the off-diagonal elements Qxy, Qxz, and Qyz, while
our way of preparing the sample crystal left some uncertainty keeping the third and all diagonal elements unchanged.
in ϑR, ϕR and �ref. Therefore, we refined these angles by mini-
The Chemical Shift Tensors
mizing for the red data set of quadrupole splittings the standard
P
N
fitted 2
1/2
deviation � = {
(1�Q,n � 1� ) /(N � 5)} , which is
(d)
cg
n=1
Q,n
Thecenterofgravityh� (� )iofadoubletd reflectsthe(neg-
n
also provided by SUPERFIT, with respect to ϑR, ϕR and �ref. It
(d)
ative) chemical shift �L � B0/B0 � � � B0/B0 plus the second-
�
turned out that �red is sensitive to changes of �ref down to 0.001
and to changes of ϑR and ϕR down to 0.01 . During the mini-
(d)
order quadrupolar shift S (�n) of a deuteron d. The partition
�
of these shifts into red and blue sets and their assignment to
one of the four rotation axes follows immediately from the anal-
mization process we identified a few data points as outliers and
discarded them. In each such case we verified that there is a plau-
sible reason for the qualification as “outlier,” e.g., overlapping
of lines.
(d)
Q
ysis of the quadrupole splittings h1� (�n)i. Because we know
(d)
(d)
the tensors Q , we may compute the shifts S (�n). For axially
symmetric tensors Q (39),
�
�
The refinement led to ϑR = 75.40 , ϕR = 16.12 , and �ref =
�
1
34.407 and brought �red down to 83 Hz. In passing we mention
"
#
�
�
2
Z Z
2
that it was by using these values of ϑR, ϕR and �ref that further
up we obtained the digits behind the decimal point of �1, �2,
and �3.
Q
1�Q(�n)
S(�n) =
1 �
.
[5]
2
4�L
QZ Z
There is a natural way to check the reliability of these num-
bers: The fitting procedure may be repeated for the blue data
set and the question is whether it converges to the same values 72.12 MHz, the maximum of S/� is 0.42 ppm. Because the
S(� ) is maximal for 1� = 0. For �-Ca(DCOO) and � =
n
Q
2
L
L
of ϑR, ϕR and �ref. When we did it we were greatly relieved to second-order quadrupole shifts are less-than-10% corrections
�
find that the minimum of �blue occured within 0.02 for the same of the chemical shifts and because the asymmetry parameters �
�
values of ϑR and ϕR and within 0.005 of the same value of �ref of both the red and the blue Q tensors are small (see Table 1),
as did that of �red. The minimum value of �blue turned out to be Eq. [5] is here a perfectly valid approximation. In Fig. 5 we
(
d)
(d)
9
7 Hz.
display the pure chemical shifts {[h� (� )i� � ]� S (� )}/�
cg
n
L
n
L
In Table 1 we list the red and blue QC tensors as they vs �. The chemical shift tensors obtained by subjecting these
are obtained after the refinement. Below it will turn out that data to SUPERFIT are listed in Table 2. The sin-curves in Fig. 5
the blue tensor (the red tensor) must be assigned to one of the correspond to the fitted tensors.
TABLE 1
The Deuteron QC Tensors in � -Ca(DCOO)
2
Principal value
Principal direction
Site 1
(kHz)
(ϑ, ϕ)




�
�
�
�
110.13 � 20.93 � 22.30
QX X= � 115.45
QYY = � 111.46
QZ Z = 226.91
eX = (51.7 , 319.6 )
�
Qblue/kHz =
� 20.93
65.60
168.75
44.53
eY = (112.8 , 30.2 )
�
�
�
22.30 168.75
eZ = (46.98 , 97.09 )
Asymmetry factor � = 0.018
Standard deviation � = 0.097 kHz
Principal value
(kHz)
Principal direction
Site 2
(ϑ, ϕ)


�
�
�
�
�
88.62
36.04 � 85.99
QX X = � 116.84
QYY = � 114.30
QZ Z = 231.14
eX = (92.3 , 317.6 )


�
Qred/kHz =
36.04 � 70.76 � 110.67
eY = (63.0 , 46.4 )
�
�
85.99 � 110.67
159.38
eZ = (27.11 , 232.11 )
Asymmetry factor � = 0.011
Standard deviation � = 0.083 kHz
Note. For convenience, both the cartesian and the principal component/principal direction representa-
ˆ
�
tions are given. Both are referenced to the aˆ , b, cˆ frame. The uncertainty of eZ is 0.035 ; that of eX and
eY is 1 .
�

2
PRECISION MEASUREMENT OF H QC AND CS TENSORS
73
TABLE 2
DISCUSSION
Principal Values and Principal Directions of the Traceless Part
of the Deuteron/Proton Chemical Shift Tensors in � -Ca(HCOO)
2
Error Considerations
Principal directionsa
The standard deviation � of the measured chemical shifts
from the best fit curves is 0.057 ppm corresponding to 4.1 Hz.
This number is also a conservative estimate of the statistical
uncertainty of both the cartesian and the principal components
of the CS tensors. The estimate is conservative because of the
large redundancy of the experimental data.
Principal values
(ppm)
(ϑ, ϕ)
Site
This work
Ref. (24)
This work
Ref. (24)
b
�
�
�
�
�
�
1
3.18 � 0.06
0.06 � 0.06
3.24 � 0.06
2.58 � 0.06
0.88 � 0.06
3.46 � 0.06
3.1 � 0.2
0.2 � 0.2
52.3 , 104.6
53 , 105
�
�
(
blue)
49.6 , 333.3
52 , 340
�
�
�
�
�
�
� 3.3 � 0.2
116.9 , 37.7
121 , 42
Provided that � is dominated by the uncertainty of the phase
�
�
�
�
2
2.7 � 0.2
92.7 , 316.2
93 , 319
1
2
correction and by the noise, we expect for � a value of � 7.1 Hz
�
�
�
�
�
�
(red)
0.7 � 0.2
158.8 , 53.1
158 , 56
�
�
= 3.55 Hz (remember, h�cg(�n)i results from averaging four in-
dividual line positions). The close matching of the expectation
with the actual standard deviation means that we have not over-
looked other significant sources of statistical errors and that cer-
tain types of potential systematic errors including the drift of B0
are negligible. The bulk magnetic susceptibility of the sample is
another potential source of systematic errors. It is not excluded
by the argument we just put forward. It has two aspects. The first
concerns the shift of the NMR line when, as viewed along B0, the
shape of the sample changes (remember, the sample is rotated).
This shift scales with the isotropic part �iso of the sample’s vol-
ume susceptibility �. For Ca-formate, �iso = 0.613 ppm (40).
This problem is essentially eliminated by shaping the sample
crystal, as we did, into cylindrical form. What remains are shifts
resulting from and proportional to the anisotropy 1� of �. For
Ca-formate, 1� is not known, but from the rather uniform dis-
tribution of the Ca ions, and from the zigzag arrangement of
the Ca(HCOO)2 units in the primitive cell (see Fig. 1), it may
safely be concluded that 1� is substantially smaller than �iso.
Shifts caused by 1� can be avoided altogether by working with
a spherical sample. For cylindrical samples the shifts are mini-
mal if the length/diameter ratio of the sample is equal to 1, which
for our sample was the case at least approximately. Although a
quantitative estimate of the influence of � on the measured CS
tensors remains difficult, we are confident that it does not exceed
� 3.4 � 0.2
69.0 , 45.2
68 , 48
Note. Difference of isotropic shifts �iso,blue � �iso,red = 0.33 ppm (this work),
0
.23 ppm (Ref. (24)).
Polar angles in the aˆ , b, cˆ frame.
a
b
ˆ
Statistical error.
0
.1 ppm.
The average standard deviation of 90 Hz of the measured dou-
blet splittings from the simulated best fit curves is also a valid
estimate of the error limits of the cartesian and principal com-
ponents of the QC tensors. These limits include statistical as
well as systematic errors. Remaining systematic errors revealed
themselves by the fact that in the process of refining the direc-
tion ϑR, ϕR of the sample’s rotation axis the scattering of the
measured splittings around the best fit curves did not become
perfectly random. The dominating statistical and systematic er-
rors result, unlike those of the chemical shifts, from the errors
of the goniometer setting and from the nonuniformly divided
toothed wheel. The size of these errors conforms with the esti-
mates given under Experimental.
FIG. 5. Chemical shifts [h�cg(�)i � �L � S(�)]/�L vs the rotation angle
�
. Top: blue data set; bottom, red data set. The full curves correspond to the
Treating, as we did implicitly, the quadrupolar Hamiltonian
HQ by first-order perturbation theory is adequate because the
doublet splittings are insensitive to second-order shifts, which
fitted CS tensors. The scatter of the data points relative to these curves indi-
cates that resonances in the spectra could be located with an uncertainty of
7 Hz.
�

7
4
SCHMITT ET AL.
do not exceed 30 Hz (see above), and because third-order terms,
The deviation � of e
Zblue 1
from b has about equal components
1
which do affect the doublet splittings, will be smaller than that parallel (� ) and perpendicular (� ) to the plane of the formate
1
k
1⊥
value by roughly a factor of QZ Z /�L � 0.003.
ion which is very nearly flat (26). While the in-plane component
�
The uncertainty of 90 Hz of the off-diagonal elements of Q (� ) � 0.8 may result from unequal charges on the oxygens of
1k
�
�
corresponds to an uncertainty of 0.015 of the unique QC tensor the formate ion, the out-of-plane component (� ) � 0.8 must
1⊥
�
direction. Adding to this number the uncertainty of 0.02 in the have its origin in interionic QC contributions. In the principal
determination of the direction of the sample’s rotation axis (see axes system of the intraionic QC tensor, it takes an off-diagonal
�
above) leads us to claim that we have measured the unique QC element of Qinterionic of roughly 4.8 kHz to push e away by 0.8
Z
tensor directions at the sites of the deuterons in �-Ca(DCOO)2 from the plane of the formate ion. We thus conclude that this
�
with an uncertainty of not more than 0.035 .
number (4.8 kHz) reflects the scale of the interionic contribu-
tions to the deuteron QC tensors in Ca-formate. By considering
the difference QZ Zred � QZ Zblue = 4.2 kHz and assigning it ex-
clusively to interionic QC contributions, we arrive at the same
The Quadrupole Coupling Tensors
We begin by comparing the principal directions eZblue and result. Unlike the argument based on � , this result is, however,
1⊥
ˆ
eZred from Table 1 with the C–D bond directions b1 = (ϑb1, ϕb1) not perfectly conclusive because a small difference of the bond
ˆ
and b2 = (ϑb2, ϕb2). From the positional parameters in (26) we lengths b and b , which is well compatible with the error limits
1
2
ˆ
�
�
ˆ
�
�
deduce b1 = (46.65 , 98.67 ) and b2 = (26.49 , 231.50 ), and of (26), can also be responsible for the observed difference of
�
�
�
1 = ^(b1, eZblue = 1.21 and �2 = ^(b2, eZred = 0.68 . It is QZ Zred and QZ Zblue
.
this result that leads us to assign Qblue to a deuteron of type 1
The asymmetry � of both Qred and Qblue must be classified as
and Qred to a deuteron of type 2. We wish to point out that for small. Nevertheless, the precision achieved in this work gives
this presentation we selected, of course, one of the deuterons of meaning to the principal directions e and e and to the differ-
X
Y
type 1 and one of type 2 such that closely matching pairs of bond ence between the “equatorial” principal components QX X and
and unique QC tensor directions are obtained.
QYY . Most important, of both Qred and Qblue it is e , and not e
Y
X
As we can be confident that eZblue and eZred are reliable within as predicted by rule (iii), which is close to the normal n of the re-
�
�
�
0.035 (=: �NMR), it depends on the precision of the neutron spective molecular plane, ^(n , e
) = 13.8 , ^(n , e ) =
1
Yblue
2
Yred
�
scattering data whether we can claim to have detected devia- 7.5 . Why is Ca-formate (and likewise cupric formate (41) and
tions of unique QC and C–D bond directions. From the error strontium formate (42), but not formic acid (43)) an exception
limits specified in (26) for the carbon and hydrogen positional to rule (iii)? The reason is, as we will argue now, the negative
parameters (296 K data set), we deduce a standard deviation charge on the oxygens of the formate ion.
�
�
n = 0.28 for both b1 and b2. We thus conclude that with a
From an extensive molecular orbital study of the EFG at the
4
(�n + �NMR) confidence there is a deviation �1 between the various deuteron sites in glycine, Davidson et al. (44) concluded
direction of the carbon–hydrogen bond b1 and the unique prin- that a point charge model (p.ch.m.) should be adequate to ac-
cipal direction eZblue of the QC tensor of deuteron 1. This result count for the direct intermolecular EFG contributions at the sites
hingescriticallyonthereliabilityofthestatederrorlimitsin(26). of deuterons. However, there is also a polarization effect whose
Therefore, we have taken the first steps to convince the project size is difficult to predict but can be, for deuterons in hydrogen
selection committee of the ILL in Grenoble, France, that a neu- bonds, three times as large as the direct effect. For deuterons
tron diffraction reexamination of the structure of �-Ca-formate, bound to carbons, it is claimed to be significantly smaller (44).
taking advantage of the availability of perdeuterated crystals, Despite all reservations, we will now consider the EFG contri-
would be a worthwhile project. For deuteron 2 and bond b2, the bution from the charge on the oxygens at the deuteron site in the
deviation �2 amounts to only 2(�n + �NMR) and we cannot say formate ion also as an intermolecular contribution and apply the
that this is a significant difference from zero.
p.ch.m. A set of point charges q , i = 1, 2, 3, . . . , at positions
i
To put the deviations �1 and �2 in perspective, we note that r = (x , y , z ) produces at the site r = (x , y , z ) of a test
i
i
i
i
k
k
k
k
the direction b1 following (26) differs from that following (27) nucleus k an electric field gradient V with components
�
�
by 0.87 . For b2, the difference is as much as 1.52 . That means
carbon–hydrogen bond directions deduced from two indepen-
dent neutron diffraction data sets can deviate from each other
as much, in fact even more, than bond and unique QC ten-
sor directions. On the other hand, as the error limits cited by
X
2
e
3(� � � )(� � � ) � (|r � r |) �
k
i
k
i
k
i
��
V�� =
(qi /e)
,
5
4
�ꢀ0
(|rk � ri |)
i
[6]
�
ˆ
ˆ
Bargouth and Will imply �n = 0.70 for both b1 and b2, the
above-quoted discrepancies between the two neutron structure where ꢀ0 is the permittivity of free space and �, � = x, y, z.
3
2
investigations are within one (combined) standard deviation for As Q = (eQ/h)V, Q�� is expressed by the same sum as
b1 and within two standard deviations for b2, and are thus no in [6] with a prefactor of 149.37 kHz if distances are mea-
reason for worrying but rather give credit to the error estimates sured in angstroms. The most critical part in applications of
of the authors.
the p.ch.m. is the choice of the charges qi /e. Here we follow,

2
PRECISION MEASUREMENT OF H QC AND CS TENSORS
75
2
very naively, a sophomore’s guess: Only the Ca ions and the fact that both Qred and Qblue violate rule (iii), as do the Q tensors
oxygens bear charges, qCa/e = +2 and, as the crystal must in cupric and strontium formate, speaks, however, against this
be neutral, qO/e = � 1/2. From the oxygens of the formate ion possibility.
which contains the test deuteron, Eq. [6] then predicts (averaged
Finally, we point out that the deuteron QC constants
2
2
over both ions) a contribution Qoxygen with principal components CQ = e qQ/h(= QZ Z ) in Ca-formate are unusually small,
3
�
20.6, 18.8, and 1.8 kHz and principal directions, in the same CQred = (154.09 � 0.06) kHz, CQblue = (151.27 � 0.06) kHz.
2
order, along the bond b, along the normal n, and perpendicular For the deuterons bound to the sp carbons listed further up, CQ
to b and n. If Qoxygen is added to a dominant tensor which is ranges from 168.4 (pyromellitic acid) to 182.2 kHz (azulene);
axially symmetric along b, we get a resulting tensor Q which the average is 178.2 kHz. Small values of CQ were also reported
indeed reflects the experimental observation that it is eY which for formic acid and for cupric and strontium formate (cf. Table 2
is close to n. However, the calculated value of 17 kHz for the in (47)), but Ca-formate is extreme. As an explanation, two rea-
difference QYY � QX X = �QZ Z is much larger than in the sons come to mind. First, formic acid and the formate ion are
2
experiment where it is 4.0 and 2.5 kHz for the blue and red ten- light units, lighter than all the molecules with sp carbons listed
sors, respectively. Does this mean that the p.ch.m. with qO/e = further up, and hence prone to large-amplitude librations. Such
�
1/2 grossly overestimates the EFG contributions from the librations reduce the time-average EFG which is accessed in an
oxygens?
experiment. A test of this proposition is measuring CQ as a func-
Not necessarily. The reason for the discrepancy between cal- tion of the temperature T . This is what Soda and Chiba did for
culated and observed values of �QZ Z may well be the assump- cupric formate (41). Lowering T from 291 to 153 K led to an in-
tion that the dominant part of the EFG is axially symmetric along crease of CQ from (155.3� 2.5) to 161 kHz, and an extrapolation
2
the bond. The carbon atom in the formate ion is sp hybridized to T → 0 K led to the conclusion that in a perfectly static lattice
as it is in many molecules with pronounced molecular planes. CQ would be 166 kHz. This is still an unusually small value for
Examples of such molecules, for which the deuteron QC tensors a deuteron bound to a carbon. Therefore, the following second
have been measured, are anthracene (10), pyromellitic acid (12), reason should also be taken into consideration: Above we argued
dimethylterephthalic acid (45), acetylsalicylic acid, also called that the nearby-pz-orbital and the charge-on-the-oxygen effect
aspirin (45), bullvalene (16), fluorene (17), azulene (46), potas- nearly cancel each other with regard to the EFG components
sium hydrogen maleate (47), and, as already mentioned, formic perpendicular to the C–D bond. By contrast, they strengthen
acid (43). All these tensors Q, a total of 24, obey rule (iii). In- each other with regard to the component parallel to the bond,
deed, an electron in a pz orbital, with a compensating positive that is, with regard to CQ. Both effects tend to lower CQ and
charge at the origin, produces at a nuclear site in the plane per- this tendency is the stronger the more charge there is on the
pendicular to the orbital an EFG whose component along the oxygens.
z axis of the orbital is larger in magnitude than that which is
perpendicular to the bond and the orbital. This is exactly what
rule (iii) states. Elementary as this explanation for rule (iii) is,
The Chemical Shift Tensors
As mentioned in the Introduction, Post measured in 1978
the proton CS tensors in �-Ca(HCOO)2 by line-narrowing
multiple-pulse NMR (23, 24). In that work, the assignment of
the measured CS tensors to the eight magnetically inequiva-
lent proton sites in the unit cell turned out to be a tremen-
dously hard problem and the assignment which Post eventually
proposed on the basis of several weak observational and ad-
ditional intuitive arguments was anything but certain, cf. (24,
p. 24, end of second paragraph). Note that in the present work
the assignment is perfectly safe. We are glad that we can now
confirm that Post’s assignment is correct. In Table 2 we included
his results in terms of the principal components (estimated er-
ror limits � 0.2 ppm) and principal directions of the traceless
part of the CS tensors. It is very satisfying that for all six prin-
cipal components the difference of the former and the present
it seems that it has never been put forward in print. In the com-
pounds listed above, the difference |QX X | � |QYY | ranges from
5
.2 (aspirin, position D2) to 20.4 kHz (bullvalene, position 3B),
the average is about 13 kHz.
In the formate ion, both effects discussed in the previous two
paragraphs are present. They lead to opposite differences of the
equatorial principal components of Q, which almost cancel each
other. This is the reason why the measured asymmetry factors
�
in �-Ca-formate are so small; see Table 1. In the light of this
discussion it even appears as fortuitous that the charge-on-the-
oxygens effect wins against the nearby-pz-orbital effect. Indeed
we cannot exclude for sure that the violation of rule (iii) in �-
Ca(DCOO)2 resultsfromintermolecularEFGcontributions. The
2
This guess leads, actually, to severe problems if we attempt getting from
the p.ch.m. an estimate of the interionic QC contributions. First, the observed result (which is three times more precise) is indeed smaller than
conformity of Q with the close C2v symmetry of the formate ions would not be
�
0.2 ppm. Likewise, the principal directions agree within the
reproducedbythecalculationswhich, second, wouldpredict QZ Zred < QZ Zblue,
contrary to what is observed. These problems disappear if we choose for the
charges qO/e = � 0.5, qCa/e = 1.4, and qC/e = 0.3. With this choice, the
p.ch.m. leads to components |Q��interionic| ranging up to 7.7 kHz with an average
of 2.4 kHz, consistent with the experimental estimate derived in the text.
expected error limits. Because the present deuteron NMR study
fully confirms Post’s proton NMR results, his discussion of the
CS tensors in �-Ca(HCOO)2 is still valid and we can restrict
ourselves here to a few remarks.

7
6
SCHMITT ET AL.
TheprincipaldirectionsofboththeblueandtheredCStensors that optimizes the symmetry of each resonance locally. Unless
�
reflect within reasonable accuracy the approximate C2v symme- the correct phase correction can be guaranteed within � 0.2 , an
try of the formate ions 1 and 2. At first glance, this is a very improvement of the signal/noise ratio beyond 100, as obtained
satisfying observation. The satisfaction is, however, a complete in this work in single shot spectra, is meaningless.
illusion because the relations between the most, intermediate,
and least shielded directions on the one hand and the C2v sym-
ACKNOWLEDGMENT
metry elements of the formate ions (twofold axis, normals n and
0
n of the mirror planes) on the other hand are different for ions
We thank the referees for critical, helpful comments.
1
and 2. While in both ions the least shielded direction is close
�
�
to n (deviation 6.4 (blue), 10.4 (red)), it is the most shielded
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�
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2
PRECISION MEASUREMENT OF H QC AND CS TENSORS
77
3
3
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