Variable NMR spin-lattice relaxation times in secondary amides: Effect of ramachandran angles on librational dynamics

Article abstract of DOI:10.1021/jp971068c

Deuterium NMR spin-lattice relaxation times (T_1Z) of N-deuterated microcrystalline secondary amides vary from less than 1 s to more than 500 s at room temperature. The main motion effecting relaxation is an out-of-plane libration of the amide, as indicated by temperature-dependent line shapes and anisotropic relaxation spectra. Over 25 amides were measured; they vary with respect to side chain sterics, hydrogen bond lengths, hydrogen bond geometry, and crystal packing. The temperature-dependent deuterium line shape and anisotropic relaxation rates indicate an out-of-plane angular deflection of approximately 10°; the angle is probably similar for the rapidly and slowly relaxing amides, while the apparent time constant for the motion probably varies dramatically. Deuterons in methylene groups on both sides of the amide group for caprylolactam and caprolactam also indicate an out-of-plane libration with relaxation rates faster than that of the amide deuteron, probably because the angular extent of the distortion is greater for the flanking α-deuteron than for the amide deuteron. Carbon relaxation measurements on lauryllactam indicate that the whole molecule librates to a comparable extent. Temperature-dependent relaxation rates for caprylolactam and caprolactam showed nonArrhenius monotonic increases in the relaxtion rates with increasing temperature, as expected for libration dynamics; furthermore the quadrupolar relaxation measurements support the assumption that the dominant spectral density contribution is above the Larmor frequency (i.e. T_1Q is longer than T_1Z). In aggregate, the data indicate that the motion effecting amide relaxation is a low-amplitude libration involving the entire molecule. Previous work on the librations of amides suggested that these librations are pronounced on the NMR time scale when the substance is near a phase transition; we report here that there is additionally a relation between the extent of libration and the structure. Comparison of the relaxation times to structures indicates that only amides with flanking alkyl groups on both sides (larger than a methyl group) exhibit extensive libration; furthermore only those amides with both flanking dihedral angles, φ{C₂C₁^-NC(=O)} and ψ{N(O=)C^-C₁′C₂′}, near -60° (～±40°) have fast spin-lattice relaxation. On the other hand, correlation between the deuterium relaxation times and hydrogen bond length nor geometry nor crystal packing was observed. Variation in the electronic structures of the conjugated amide groups was indirectly probed by measuring the chemical shift anisotropy of the amide carbonyl carbon, the deuterium quadrupolar coupling constant, and vibrational frequencies. These parameters did not vary dramatically, indicating that the electronic structure is not strongly variable; the modest variation did not correlate with deuterium relaxation rates. The chemical shift tensor elements were δ₁₁ = 91.4 ±5, δ₂₂ = 185 ± 8, and δ₃₃ = 245 ± 3 ppm, the quadrupolar coupling constant and its anisotropy were 203 ± 10 kHz and 0.15 ± 0.02, the NH stretch frequency was 3300 ± 42 cm^-1, and the carbonyl stretch frequency was 1644 ± 25 cm^-1. We suggest a model in which the shape of the local potential associated with flanking alkyl groups leads to "overdamping" of the amide librational mode and generates slower (nanosecond) components in the vibrational frequency spectrum.

Full text of DOI:10.1021/jp971068c

6248
J. Phys. Chem. B 1998, 102, 6248-6259
Variable NMR Spin-Lattice Relaxation Times in Secondary Amides: Effect of
Ramachandran Angles on Librational Dynamics†
John C. Williams‡ and Ann E. McDermott*
Department of Chemistry, Columbia UniVersity, New York, New York 10027
ReceiVed: March 25, 1997X
Deuterium NMR spin-lattice relaxation times (T1Z) of N-deuterated microcrystalline secondary amides vary
from less than 1 s to more than 500 s at room temperature. The main motion effecting relaxation is an
out-of-plane libration of the amide, as indicated by temperature-dependent line shapes and anisotropic relaxation
spectra. Over 25 amides were measured; they vary with respect to side chain sterics, hydrogen bond lengths,
hydrogen bond geometry, and crystal packing. The temperature-dependent deuterium line shape and anisotropic
relaxation rates indicate an out-of-plane angular deflection of approximately 10°; the angle is probably similar
for the rapidly and slowly relaxing amides, while the apparent time constant for the motion probably varies
dramatically. Deuterons in methylene groups on both sides of the amide group for caprylolactam and
caprolactam also indicate an out-of-plane libration with relaxation rates faster than that of the amide deuteron,
probably because the angular extent of the distortion is greater for the flanking R-deuteron than for the amide
deuteron. Carbon relaxation measurements on lauryllactam indicate that the whole molecule librates to a
comparable extent. Temperature-dependent relaxation rates for caprylolactam and caprolactam showed non-
Arrhenius monotonic increases in the relaxtion rates with increasing temperature, as expected for libration
dynamics; furthermore the quadrupolar relaxation measurements support the assumption that the dominant
spectral density contribution is above the Larmor frequency (i.e. T1Q is longer than T1Z). In aggregate, the
data indicate that the motion effecting amide relaxation is a low-amplitude libration involving the entire
molecule. Previous work on the librations of amides suggested that these librations are pronounced on the
NMR time scale when the substance is near a phase transition; we report here that there is additionally a
relation between the extent of libration and the structure. Comparison of the relaxation times to structures
indicates that only amides with flanking alkyl groups on both sides (larger than a methyl group) exhibit
-
extensive libration; furthermore only those amides with both flanking dihedral angles, φ {C2C1 NC(dO)}
and ψ {N(Od)C-C1′C2′}, near -60° (∼(40°) have fast spin-lattice relaxation. On the other hand, correlation
between the deuterium relaxation times and hydrogen bond length nor geometry nor crystal packing was
observed. Variation in the electronic structures of the conjugated amide groups was indirectly probed by
measuring the chemical shift anisotropy of the amide carbonyl carbon, the deuterium quadrupolar coupling
constant, and vibrational frequencies. These parameters did not vary dramatically, indicating that the electronic
structure is not strongly variable; the modest variation did not correlate with deuterium relaxation rates. The
chemical shift tensor elements were δ11 ) 91.4 ( 5, δ22 ) 185 ( 8, and δ33 ) 245 ( 3 ppm, the quadrupolar
coupling constant and its anisotropy were 203 ( 10 kHz and 0.15 ( 0.02, the NH stretch frequency was
3300 ( 42 cm-1, and the carbonyl stretch frequency was 1644 ( 25 cm-1. We suggest a model in which the
shape of the local potential associated with flanking alkyl groups leads to “overdamping” of the amide librational
mode and generates slower (nanosecond) components in the vibrational frequency spectrum.
Introduction
postulated to aid chemical reactivity in solid state systems by
providing directional collisions between the reactants, a phe-
nomenon termed phonon assistance,4 which has also been
discussed in enzyme catalysis.5 For the amide backbone in
proteins one might expect a rather direct connection between
low-angle fluctuations and control of function such as binding
or catalysis through electrostatic terms; the charge-separated
resonance structure of the amide produces a substantial dipole
moment and any motions of the amide modify the frequency
dependence of the local dielectric constant of proteins in a
substantial way. Systematic studies identifying or characterizing
low-frequency motions and their dependence on structure are
still lacking.
Our understanding of librations of chemical and biochemical
systems and their consequences to chemistry and macromo-
lecular function is far from complete. The existence of low-
frequency modes in proteins has been used to explain hydrogen
exchange experiments in hemoglobin1 and trypsin2 and oxygen
diffusion in myoglobin.3 Low-frequency motions have been
† Acknowledgment is made to the Donors of the Petroleum Research
Fund, administered by the American Chemical Society, for partial support
of this research. A.E.M. acknowledges support from Kanagawa Academy
of Science and Technology and National Institutes of Health GM50291.
J.C.W. acknowledges support from NIH Biophysics training grant GM98281.
‡ Currently at the European Molecular Biology Lab, Meyerhoffstrasse
1, D69012 Heidelberg, Germany.
This study reports solid state NMR measurements of libra-
tional dynamics of secondary amides. Amides are known to
* To whom correspondence should be addressed.
X Abstract published in AdVance ACS Abstracts, July 1, 1997.
S1089-5647(97)01068-7 CCC: $15.00 © 1998 American Chemical Society
Published on Web 07/21/1998

Spin-Lattice Relaxation Times in Secondary Amides
J. Phys. Chem. B, Vol. 102, No. 32, 1998 6249
librate out of plane in a range of 8-15° in small molecules and
in proteins. For isolated molecules, the frequency of out-of-
plane librational modes would be expected to be in the far
infrared region (10-200 cm-1), much too high to provide an
NMR relaxation mechanism and probably too fast to impact
function of biopolymers. In the condensed phase and in
macromolecules in particular, librational modes with amide
rocking character can be observed on dramatically slower time
scales. The lower frequencies in the condensed phase can be
explained in a variety of ways. Formation of hydrogen bonds
to adjacent groups could restrict the extent of excursion of the
amide from its equilibrium position; conformations of adjacent
methylene groups and attached bulky side chains may also affect
the time scale, and coupling of modes from neighboring groups
(for example electrostatically) might induce collective motions
and lower the frequency.
mol) was placed in the dropping funnel and added over a period
of 15 min to the amine/ether solution. A white precipitate was
seen after the first drop. The solvent was removed by rotary
evaporation, and the solid was dissolved in methylene chloride.
The methylene chloride was then washed several times with a
5% sodium bicarbonate solution. The organic layer was
separated, pooled, and evaporated under reduced pressure. The
resulting white crystalline material was then recrystallized
several times with methanol. The NMR and IR confirmed the
formation of an amide. The final yield was 5.1 g or 64% (0.044
mol). In certain cases, a light orange solution of viscous
material resulted, and sublimation under reduced pressure and
heat provided pure material. Solution 1H NMR, 13C solid state
NMR, and infrared measurements indicated the desired product.
Polyamide-4,6 was generously provided to us from Dr. I. Vulic’,
DSM.21
Previous studies of libration in proteins involved a wide
variety of experimental approaches, mainly sensitive to fast time
scales. Raman spectroscopy,6,7 inelastic neutron scattering
techniques,8 dielectric relaxation,9 and acoustical methods have
provided information about crystalline dynamics, particularly
for frequencies in the giga- to tetrahertz regime. Computational
studies such as molecular dynamics and normal mode calcula-
tions have also provided detailed information about these
modes,10,11 and they too have mostly concentrated on motions
in the tens of nanoseconds to femtosecond regime. Solid state
NMR methods can extend these measurements to the giga- to
kilohertz regime.12-14 SSNMR methods can be useful for
detailed studies of biopolymers when specific isotopic labeling
is possible.
An unexpected variability in the librational dynamics of the
amide plane among 25 secondary amides is reported herein,
which seems to be related to the presence of torsional angles
of the flanking alkyl groups. Compilation of the relaxation times
of the amidic deuteron with many other properties possibly
related to hyrogen bond strength fails to yield a correlation with
motion time scale; these include the intramolecular conforma-
tion, hydrogen bond length and geometry, crystal packing
geometry, and spectroscopic signatures; i.e., the quadrupolar
coupling constant of the deuteron, its anistropy,15-18 13C chemi-
cal shift anisotropy, and infrared frequencies. A model for the
motion is discussed in which the relation between flanking
torsional angles and time scale of fluctuation is addressed.
The amidic proton of each sample was exchanged in D2O or
ethanol-d. The exchanged samples were recrystallized in the
same solvent as reported in the crystal structure, and the IR of
each amide indicated better than 75% deuteration in each case.
Preparation of r-Deuterio Compounds. The R-deuterated
compounds were prepared from the parent cycloketone and
hydroxylamine hydrochloride via the Beckman rearrangement.
The following procedure resembles closely that used for the
R-deuterated caprolactam. The R-deuterons of the cyclooctoc-
tanone were exchanged twice in NaOD/CH3OD/D2O. The
exchange was monitored via solution 1H NMR {RCH first
exchange (CDCl3, 88% D) δ ppm ) 2.32 triplet of triplets;
second exchange, undetected}. Hydroxylamine hydrochloride
was exchanged twice in D2O and dried by rotary evaporation.
The exchanged hydroxylamine (1.89 g) was added to 5 mL of
D2O and slowly neutralized with anhydrous potassium acetate
(2.68 g). To this solution was added the exchanged cyclooc-
tanone in 8 mL of methanol-d. The solution was refluxed with
gentle stirring for 3 h. The solution was then cooled and
extracted three times with chloroform. The chloroform extract
was dried by rotary evaporation, producing a white crystalline
material. Proton NMR of a protonated control at this point
indicated the formation of the oxime due to the splitting of one
of the R-carbon’s protons {RCH ) triplet; δ ) 2.36, 2.40, 2.43
and 2.19, 2.22, 2.25)}. A solution of hydrosulfuric acid and
water (10:2 v/v) (3 mL) was added to the crystalline material
and gently heated until a deep orange solution resulted. The
solution was neutralized with sodium hydroxide and extracted
three times with chloroform. The chloroform layer was washed
with sodium bicarbonate twice, dried, and evaporated. NMR
of the extremely viscous material (protonated trial) was identical
to the spectrum of the purchased product. NMR of the
deuterated compound showed similar shifts in the â-carbon as
well as the amide proton (vb 5.8 ppm). Sublimation three times
produced the pure material. The overall yield was 50%.
Experimental Section
Sample Preparation. Samples were either purchased from
Aldrich or Janssen or synthesized. Several amides were
synthesized (N,N′-heptamethylene dibenzamide, N,N′-hexa-
methylenediacetamide, N,N′-hexamethylenedibenzamide, N,N′-
heptamethylenedipropionamide, N,N′-octamethylenedibenza-
mide, N-tert-butylacetamide, N-tert-butylpropionamide, N,N-di-
n-butyladipamide, and N,N-isopropyladipamide) by reacting 1
equiv of the appropriate acid chloride with 1 equiv of the
appropriate amine in a solution of triethylamine and methylene
chloride or ether and cooling in a dry ice/acetone bath.19,20
N-tert-butylacetamide is a typical example. An oven-dried,
three-neck round bottom flask was equipped with an equal
pressure dropping funnel and a stir bar with dry nitrogen gas
purge. A solution of tert-butylamine (14.5 mL/0.138 mol) and
ether (20 mL) was placed in the round bottom flask and cooled
in a dry ice/acetone bath. When ether was used as a solvent,
tert-butylamine was added in 2-fold excess to neutralize the
chloride. For other syntheses, freshly distilled triethylamine was
used to neutralize the chloride. The acetyl chloride (4.9/0.069
NMR Measurements. Spectra were collected on a Che-
magnetics CMX400 equipped with a broad-line deuterium probe
and a 7 mm CPMAS probe. The deuterium line shapes were
collected with a standard quadrupolar echo sequence. The
anisotropy experiments were collected with an inversion pulse
followed by a variable delay time and detected with the
quadrupolar echo sequence. The π/2 pulse length was set to
2.0 µs, and the FID was sampled at 2 MHz with a total length
of 1024 complex points. The relaxation delay was ap-
proximately 5T1 for each sample (varying from 3 to 600 s). All
spectra were processed with 2 kHz of line broadening with the
exception of the variable-temperature line shape study, wherein
subtle changes in line shapes were studied without line

6250 J. Phys. Chem. B, Vol. 102, No. 32, 1998
Williams and McDermott
TABLE 1: Deuterium Spin-Lattice Relaxation Data and the 13C CSA Values
compound
2H T (s)
eEqQ (kHz)
η
δ
11 (ppm)
88
δ
22 (ppm)
197
δ
33 (ppm)
246
(1) acetamide
(2) urea
115
50
64
290
>200
>300
>300
>600
1.2
197
211
177
0.12
0.11
0.16
(3) caprolactam
(4) glycine anhydride
(5) homoazaadamantone
(6) 2-pyridone
97
82
102
84
200
182
189
172
175
249
244
242
237
248
210
189
194
203
0.18
0.19
0.16
0.17
(7) acetanalide
(8) benzanilide
90
(9) caprylolactam
(10) caprylolactam‚HCl
95
82
90
193
201
190
249
262
238
1.2
1.5
45
(11) cyclic monomer N-6,6
(12) N,N′-heptamethylenedibenzamide
(13) N,N′-hexamethylenediacetamide
(14) N,N′-hexamethylenedibenzamide
(15) N,N′-hexamethylenedipropionamide
(16) N-methylbenzamide
(17) N,N′-octamethylenedibenzamide
(18) polyalanine
172
>300
166
288
>300
1.0
85
181
186
237
240
100
95
92
94
94
92
88
93
88
96
86
178
189
183
174
181
184
175
183
183
174
243
243
250
246
246
248
245
249
244
243
(19) polyamide 4,6
4.3
(20) N-tert-butylpropamide
(21) N-(1-adamantyl)acetamide
(22) lauryllactam
(23) N,N′-hexamethylenedihexamide
(24) N-tert-butylacetamide
(25) N,N′-di-n-butyladipamide
(26) N,N′-diisopropyladipamide
(27) N,N′-tetramethylenedi-butylamide
(28) N-acetylglycine
∼7
193
205
0.134
0.154
27
0.43
144
1.4
7.2
3.5
100
1.0
179
0.136
(29) polybenzyl glutamate
broadening. T1 values reported in Table 1 were obtained by
fitting the integrated intensity of the Fourier transform of the
FID to I(t) ) I0(1 - 2 exp[-t/T1]).
spectra (Vide infra). Furthermore, we do not know the extent
to which the principal normal mode for the motion involves
flanking alkyl groups or neighboring molecules; this extent could
vary from one rapidly relaxing compound to another. Nonethe-
less, we looked for correlations between T1 and structure; the
interpretation of any correlation that we find has an implicit
assumption of an essentially common relaxation mode.
The 13C CPMAS spectra were collected with decoupling
powers over 50 kHz and contact times of 3 ms. Spectra from
three spinning speeds were used to determine the isotropic shift
and to estimate the error in the CSA measurements. The tensor
elements were extracted by Herzfeld-Berger analysis.22
Several general observations can be made about the structures
and packing of the amides. Amide hydrogen bond donors and
acceptors are invariably satisfied in crystalline structures.23-27
The hydrogen bond length varies over the range 2.7-3.1 Å
(NH‚‚‚OdC), and the angle (NH‚‚‚OdC) is more widely
variable (110-180°). A study of over 1400 CO‚‚‚HN hydrogen
bonds characterized by X-ray crystallography as well as several
by neutron diffraction did not find a strong preference in the
intermolecular NH‚‚‚OC bond angle.27 Probably the reason for
the variation and weak preference is that the hydrogen bond
geometry is dictated by other considerations. There are distinct
geometries that are known for cis and trans amides (Figure 1).24
Cis amides often pack as cyclic dimers, and then these dimers
are propagated by a translation element to complete the crystal
(e.g., caprolactam) or utilize a local 2-fold screw element, which
results in an infinite helix (e.g., R-pyridone) (Figure 1). In this
case the NH‚‚‚OC bond angles are typically 120°. This packing
arrangement is impossible for trans amides. For trans amides,
the amide often packs using a simple translation (e.g., capry-
lolactam (Figure 1) or a 2-fold glide (e.g., N-methylbenzamide)
(Figure 1); the NH‚‚‚OC bond angle is generally 140-180°.
The side chains for cis amides prohibit the formation of a
packing arrangement with a simple 2-fold glide. We use these
motif designations to describe the amides in this study; however
no relation between packing and relaxation behavior was seen.
Results and Discussion
2
Correlation of H T1Z and Structure. We have measured
the deuterium spin-lattice relaxation rates of approximately 25
amides, which range from less than 1 s to more than 600 s (Table
1) most of which have been characterized by X-ray crystal-
lography (Table 2). Although anisotropic relaxation is an
especially useful method in discriminating different dynamical
models (Vide infra), we report the relaxation times based upon
the intensity of the first point of the FID (related to the integrated
intensity of the spectrum) for many of these compounds because
of their long spin-lattice relaxation times and associated poor
signal-to-noise ratios. In order to analyze the variation in the
deuterium relaxation rates of these crystalline amides, we
selected amides that vary in intramolecular conformation,
hydrogen bond geometry, crystalline packing arrangement, and
side chain moiety.
In studying the correlation between structure and relaxation
rate, we are making an implicit assumption that the motion that
drives relaxation is common throughout the database. The
relaxation mechanism is common among all of the samples
studied, to the extent that all of the cases with efficient relaxation
are driven by a librational process. On the other hand, the
motion varies from one compound to another in terms of the
extent of anisotropy; in other words, some compounds relax
through purely out-of-plane libration, while others enact a more
isotropic motion such as “wobbling in a cone” or “wobbling
on the surface of a cone”, as evidenced by their T1 anisotropy
The only correlation we have detected between the amide
deuteron relaxation rate (2H T1Z) and the structure of the
compound involves the alkyl groups adjacent to the amide. Only
amides with flanking alkyl groups larger than a methyl group

Spin-Lattice Relaxation Times in Secondary Amides
J. Phys. Chem. B, Vol. 102, No. 32, 1998 6251
Figure 1. Different packing forms common for cis and trans amides,
including the compounds used in this study. Each packing form places
geometric restrictions on the hydrogen bond.
exhibit librational dynamics on the NMR time scale. Further-
more there is a rough relation between the dihedral angles in
these flanking groups and the presence of “slow” librational
dynamics. A φ/ψ map similar to those used to describe peptide
conformation (Ramachadran plot) is shown in Figure 2. (Note
that unlike the conventional plot used to describe protein
structure we are plotting the two angles flanking an individual
amide.) Compounds with a conformation near φ ) -60° and
ψ ) -60° include caprylolactam, polyamide-4,6, the cyclic
monomer, polyalanine, and PBG; these compounds all have fast
2H T1Z values (Table 2). The 2D map exhibits an “island” of
fast relaxing amides. Compounds with this conformation are
also characterized by gauche eclipsing interactions between the
amide and the protons of the flanking methylene. For the
polyamide-4,6, the cyclic monomer, and caprylolactam the
formation of the ring forces the adjacent methylene groups to
take on a gauche conformation. In the case of polyalanine and
PBG, the formation of the hydrogen bond to the i+4 and i-4
amides necessitates these dihedral angles. For acyclic amides
such as hexamethylene dibenzamide this restriction is lifted and
the methylene chain can take on an all-trans conformation.
Several of the compounds that are not displayed on the φ/ψ
map are branched at the methylene group such that the dihedral
angle is not well-defined. For instance, tert-butylpropionamide,
which has a “medium” value T1 of 7.2 s, cannot be defined
since the group adjacent to the amide nitrogen is a tertiary butyl
group. The other dihedral, NC(dO)-CC, is -147°, which is
similar to the fast relaxing amides. Dihedral angles for others
such as N-methylbenzamide or hexamethylenediacetamide can-
not be defined since no fourth main chain atom exists (e.g.
NHC(dO)CH3); as mentioned above, these compounds, with
only a methyl group on one side, exhibit a long relaxation time.
Some amides in our study that lack crystallographic charac-
terization also have a short relaxation time. The spin-lattice
relaxation times for N,N′-di-n-butyladipamide, N,N′-di-n-pro-
Figure 2. (top) φ/ψ angles plotted as a function of the logarithm of
deuterium longitudinal relaxation time. The numbers adjacent to the
boxes refer to the compound given in Table 2. The points with short
spin-lattice relaxation times are clustered about φ/ψ angles of an
R-helix, {-60°,-60°}. Some of the amides with long spin-lattice
relaxation times occupy regions of φ/ψ that are normally inaccessible
for peptides with the exception of point 12, which occupies the region
assigned to the â-sheet. (bottom) Projection of the values onto the
φ/ψ plane: the solid circles represent compounds with a relaxation
time less than 5 s (most ∼1 s), and the solid squares represent
compounds with a relaxation time longer than 50 s (most of which are
greater than 100 s). This figure represents the definition of the
Ramachadran angles used in this study and typically used to describe
the secondary structure of peptides.
pyladipamide, and nylon-4,4 are 1.4, 7.2, and 1 s, respectively.
The crystal structure of a similar compound, N,N′-diethyladi-
pamide, shows that the dihedral angle adjacent to the carbonyl
is also gauche;28 if our compounds pack in the same manner,
they would agree with the proposed trend between dihedral
angles and relaxation times. Nylon 66 exhibits a short spin-
lattice relaxation for both the methylene and the amide29 and
has flanking dihedral angles in an all-trans conformation. This
compound would break the general rule we are proposing about
the relation between flanking dihedral angles and relaxation rate,
unless gauche defects occur in large population near the phase
transition. We report herein data on a compound similar to
nylon 66, N,N′-hexamethylenehexamide, but with the amides
reversed {e.g. C(dO)N vs NC(dO)}, which has a long spin-
lattice relaxation rate.

6252 J. Phys. Chem. B, Vol. 102, No. 32, 1998
Williams and McDermott
TABLE 2: Crystallographic Information for the Compounds Used in This Study
IR bands
torsional angles
compound
NOC N‚‚‚O CO NH CCNC ψ CNCC ω NCCC φ packing
ref
(1) acetamide
(2) urea
(3) caprolactam
(4) glycine anhydride
(5) homoazaadamantone
(6) 2-pyridone
(7) acetanalide
(8) benzanilide
(9) caprylolactam
121/120
147/167
121
122
119
136
139
161
174
2.99
3.02
2.90
2.85
2.90
2.77
2.94
3.11
2.86
1607
3D array [Hamilton, 1965]
3D array [Donahue, 1940]
1656 3223
1705 3193
1657 3196
1639
67.8
-1.3
-4.1
1.3
-63.2 cyclic
[Winkler, 1975a]
[Degeilh, 1959]
[Symersky, 1986]
-1.2 cyclic
-3.5
cyclic
0
-1
0
P2 screw [Penfold, 1953]
P2 screw [Brown, 1966]
1642 3298 -160
177
1646 3347
1634 3319
1685
-31.3
∼180
31.6 P2 screw [Kashino, 1979]
-88.8 trans [Winkler, 1975b]
-90.4
148.4
-171.4
175.5
(10) caprylolactam‚HCl
(11) cyclic monomer N-6,6
(12) N,N′-heptamethylenedibenzamide
86.6
-101.1 P2 screw [Winkler, 1975b]
140
151
2.88
2.88
1633 1634 -145.3
3323 1628
75.3 P2 screw [Northolt, 1968]
92.6
-105.1
172.2
155.4 P2 glide
145.8
[Brisson, 1982]
-172.2
-178.6
-178.9
-176.9
179.1
179.1
179.
(13) N,N′-hexamethylenediacetamide
(14) N,N′-hexamethylenedibenzamide
(15) N,N′-hexamethylenedipropionamide 174
(16) N-methylbenzamide
(17) N,N′-octamethylenedibenzamide
(18) polyalanine
175
165
2.88
3.01
2.95
2.92
3.01
2.86
2.87
2.98
1625 3309 -168.1
trans
[Bailey, 1955]
[Pineault, 1983]
[Jensen, 1957]
[Leisoriwitz, 1978]
[Pineault, 1983b]
[Arnott, 1966]
[Vulic´, 1994]
1630 3322
1634 3312
1631 3333
171.2
168.6
-152.5 trans
-169.5 P2 glide
-14.3 P2 glide
-152.5 trans
-54.
140
164
164
167
160
-170.9
-57.
-74.
1645 3282
1630 3311
(19) polyamide 4,6
167.1
-174
-60.8
(20) N-tert-butylpropionamide
(21) N-(1-adamantyl)acetamide
(22) lauryllactam
(23) N,N′-hexamethylenedihexamide
(24) N-tert-butylacetamide
1646 3319 -147
1636 3324
1641 3310
1634 3306
1648 3290
1669
[this study]
(25) N,N′-di-n-butyladipamide
(26) N,N′-diisopropyladipamide
(27) N,N′-tetramethylenedibutylamide
trans
trans
1626 3303
1638 3306
In order to study the effect of the alkyl group, a homologous
series with the same packing arrangement was studied: hex-
amethylenediacetamide, hexamethylenedipropionamide, hex-
amethylenedihexamide, and hexamethylenedibenzamide {RC-
(dO)NH(CH2)6NHC(dO)-R, where R ) methyl, ethyl,
hexamethyl, phenyl groups}. All of these compounds exhibit
long relaxation rates. In another series of compounds with
similar packing the alkyl groups vary enormously, but the
relaxation rates are consistently very long: R-C(dO)NH-R′:
acetanalide, where R ) methyl group and R′ ) phenyl group;
N-methylbenzamide, where R ) phenyl group and R′ ) methyl
group; and benzanalide, where R ) R′ ) phenyl group.
Strain incurred by closing the ring in lactams causes the amide
unit to become nonplanar. We considered the possibility that
such a strain would cause variation in the electronic structure
and therefore a variation in conformational dynamics. This
deviation is expressed in terms of a dihedral angle, ω ) CN-
C(dO)C, where values are either greater than 0° for cis amides
or less than 180° for trans amides.30 The deformation defined
here is the magnitude from either trans, |ωd| ) |180 - ω|, or
cis, |ωd| ) |0 - ω|. For caprylolactam this deformation is
considerable, |ωd| ) 33°, a particularly large deviation. (This
correlation is available in the Supporting Information, Figure
1.) Although there is a somewhat greater range in nonplanar
distortion for the amides with a fast deuterium T1 as compared
to the amides with a slow deuterium T1, the correlation is
unconvincing.
relaxation time of 167 s). The packing arrangement fails to
correlate with the deuterium relaxation rates. (See Figure 4-3
of ref 56.) The density and the Debye-Waller thermal factors
(data not shown) derived from the crystal structure also did not
correlate with the relaxation rates.
Chemical Shift Tensors, QCC, and IR. Hydrogen bonding
can have a dramatic effect on the electronic structure of the
hydrogen bond donor and acceptor. This is evidenced by band
shifts in IR frequencies, changes in UV-vis absorption, and
NMR chemical shifts. To test whether the variable dynamics
responsible for the variable deuterium relaxation times are
associated with changes in electronic structures of the amides,
we have measured the 13C chemical shift anisotropy of the
carbonyl, the IR spectra, and the quadrupolar coupling constant
(QCC) of most of these amides, and we have compared these
spectroscopic signatures to the structures as well as to the
relaxation rates. In response to hydrogen-bonding interactions,
the oxygen of an amide group purportedly becomes more
negatively charged and the CN bond more double bond-like;
these features have been invoked to explain the trends in IR
spectra, NMR spectra, X-ray lengths, and cis-trans rotation
barriers. This hypothesis, however, is not borne out by our data.
Infrared spectra for each compound offer evidence that the
resonance structures are not substantially variable for most of
the secondary compounds and are not strongly affected by ring
strain. The NH stretch and the amide I stretch (mostly
associated with the carbonyl) are reported in Table 2. The
average value of the carbonyl frequency is ∼1645 ( 23 cm-1
,
From the data compiled in Tables 1 and 2, we can rule out
other structural properties as determining the deuterium relax-
ation rates and associated dynamics: there is no correlation
between the deuteron relaxation rate and the amide conformation
(cis vs trans), nor for hydrogen bond length, hydrogen bond
angle, or packing form. Typically the trans amides relax faster
than the cis amides; however, there are exceptions (e.g.,
hexamethylenediacetamide, a trans amide with a spin-lattice
and the NH stretch is 3254 ( 80 cm-1, with no significant
correlation with the ring strain (data not shown). IR bond
stretches were compared to the CO, CN, and N‚‚‚O bond
lengths, revealing no correlation. The correlation between the
intermolecular N‚‚‚O bond length and the CO bond length or
between the CN bond length and the CO bond length was also
weak (see Figure 4-11 of ref 56).

Spin-Lattice Relaxation Times in Secondary Amides
J. Phys. Chem. B, Vol. 102, No. 32, 1998 6253
The chemical tensor elements of the amide carbonyl are
shown in Figure 3; δ22 is along the CdO bond direction, δ33 is
out of the amide plane, and δ11 is normal to both and in-plane,
approximately 30° to the CN bond.31 Asakawa et al. have
measured the chemical shift tensors for six alanine-containing
peptides and calculated the tensor for canonical peptide con-
formations, the â-sheet, and R-helix.32 As the hydrogen bond
length, RN‚‚‚O, increases, the δ33 element shifts downfield and
the δ22 element shifts upfield, but little change in the δ11 element
was reported, both for experimental and calculated shifts. In
contrast, previous solid state NMR studies of the carboxyl
group33 show that the principal elements of the chemical shift
tensor, δ11, δ22, and δ33, are very sensitive to the length and (in
the case of carboxylic acids) the geometry of the hydrogen bond.
In the carboxylic acid study, δ11 showed little variation, whereas
δ33 correlated well with whether the carboxyl group was
protonated or deprotonated and δ22 correlated well with the
hydrogen bond strength.
in the δ11 and δ22 elements when compared to the neutral
compound. These changes are most likely due to the protona-
tion of the carbonyl. Acetamide also exhibits a relatively
downfield δ22 shift and has two hyrogen bonds for the carbonyl
(whereas most accept only one). Considerable changes in the
CN and CO bond lengths are observed for the neutral and the
HCl form of caprylolactam. The average bond lengths for the
CN and CO bonds for all compounds in this study are 1.247 Å
((0.022) and 1.338 Å ((0.015). The CN and CO bond lengths
are 1.34 and 1.24 Å for caprylolactam and 1.31 and 1.30 Å for
the HCl salt of caprylolactam, respectively;36 a 7° change in
the O-C-N bond angle is also seen from the two compounds.
Thus, the spectroscopic indicators certainly are sensitive to these
more dramatic geometric changes upon full protonation, but the
effects of weak hydrogen bonding are apparently masked by
other environmental and internal variations in the compounds.
The quadrupolar coupling constant, QCC ) e2qQ/h, and its
anisotropy, η, were also measured. These values probe the
electric field gradient experienced at the deuterium nucleus and
are sensitive indicators of the hydrogen bond strength. An
empirical relation shows that the coupling is related to the
ND‚‚‚O distance by QCC ) A - B/r3, where A ) 282 and B )
-572 and r is the distance between the deuteron and oxygen.18
The values for both the coupling constant and the quadrupolar
anisotropy are presented in Table 1. The quadrupolar coupling
constants that were measured showed a modest variation, as
expected (data not shown). However, none of the crystal
structures of the amides in our study was solved using neutron
methods, so the lengths of the NH distance as well as the H‚‚‚O
distance were not amenable to detailed analyses.
The tensor elements for the amides reported herein are given
in Table 1 and are within the range previously reported in the
literature. Several of the tensors were not measured due to
prohibitively long proton relaxation (which affects the rate of
data acquisition for CPMAS studies). It is interesting that the
proton relaxation times reflected the same trend observed for
the deuterium T1, despite the presence of methyl groups in some
(such as N-methylbenzamide, which had a proton T1 on the order
of 2 min). Spin-lattice relaxation rates for the protons,
however, were not quantitatively measured in general and are
not reported.
In our database, the 13C CSA were only poorly correlated
with the bond lengths and other measures of the hydrogen bond;
the hydrogen bond strength in this series of compounds is
probably not variable enough to show the trends clearly. In
particular, no correlation was observed between the 13C CSA
tensor elements and the NH‚‚‚OC bond angle or N‚‚‚O bond
length nor with the out-of-phase deformation caused by ring
strain. Furthermore, there is none between the deuterium T1
and the chemical shift elements (data not shown). Moreover,
there is no correlation between the chemical shift tensor elements
and the infrared frequencies. (See Figure 4-10 of ref 56.) There
is, however, a previously unreported correlation between δ22
and δ33 and the dihedral angle ψ which might be relevant for
protein studies.
The modest variation of the 13C CSA elements in response
to structural properties may be due to the limited hydrogen bond
range, ∼2.7 to 2.9 Å. The limited range has been noted
previously; a study of the Coulombic force associated with each
of these different packing and hydrogen bond geometries
indicate that the contribution of the Coulomb energy to the
hydrogen bond strength is not strongly affected by modest
changes in the length or the angle.25 It has also been shown
that the low-field shift in 13C solution NMR spectra due to
carbonyl protonation is compensated by the upfield shift due
to the formation of NH hydrogen bond.34 Such a compensation
may be responsible for the lack of variation in the chemical
shift tensor elements for weak hydrogen bonds. A detailed
comparison of the chemical shift tensors for 15N and 13C of
acetanalide (which forms hydrogen bonds) and N-methylacet-
analide (which cannot form hydrogen bonds) also showed little
differences in the tensor elements of the carbonyl carbon (δ11
) 248 vs 243, δ22 ) 175 and 175, and δ33 ) 90 vs 93 ppm,
respectively).35
These measurements indicate that the carbonyl CSA, IR, and
quadrupolar coupling constants for secondary amides vary only
to a subtle extent, and presumably the electronic structure varies
rather little as well, with the exception of the HCl salt. A much
more dramatic trend in chemical shift tensor elements for
carboxyl groups has been observed previously in response to
hydrogen bonding. Our conclusion is that the variation in
librational dynamics is unlikely to result from variation in
electronic properties associated with hydrogen bonding.
Librational Dynamics. We present a model for the type of
motion responsible for the relaxation based upon our data and
previous discussion in the literature. Some characteristics of
the motion are directly available from the data, for example
the direction of the libration, the approximate angle, and the
fact that it is not an activated process, and these characteristics
will be discussed first. To obtain other characteristics such as
the effective correlation time, we must assume a model for the
motion. Models for librational dynamics and typical time scales
are discussed in the following section, based upon previous
literature. Finally, we offer an interpretation for the connection
between the flanking alkyl groups and the librational charac-
teristics.
The direction of the libration and the approximate excursion
angle are two characteristics directly available from the data.
The singularities observed in a static deuterium spectrum
correspond to deuteron at specific angles with respect to the
applied field; changes in the positions of these singularities with
temperature are indicative of the type of molecular motion
experienced by the deuteron (Figure 4). Single-crystal studies
of urea and N-acetylglycine have shown that the Vzz component
is along the N-D bond, the Vyy component is normal to the
amide plane, and the Vxx is in the amide plane, perpendicular
to the remaining components.37,38 All amides in this study
A substantial change in the tensor elements was observed
for the HCl salt of caprylolactam. A downfield shift was seen

6254 J. Phys. Chem. B, Vol. 102, No. 32, 1998
Williams and McDermott
Figure 4. Line shapes of lauryllactam taken at -50 and 120 °C and
the difference (scaled by the Boltzmann factor). The singularities are
marked Vxx, Vyy, Vzz, corresponding to the principle axes the quadrupole
tensor. The singularity labeled Vzz corresponds to all amides with N-D
bonds aligned along the applied field. The singularity labeled Vxx
corresponds to all amides with N-D bonds and vectors normal to the
amide plane perpendicular to the applied field. Finally, the component
marked Vyy corresponds to all amides both with the N-D bonds
perpendicular to the applied field and with vectors normal of the amide
plane parallel to the applied field. The spectral intensity in the
difference spectra is mostly associated with the y-component of the
electric field gradient and indicates an out-of-plane libration. By
increasing the temperature, the angular excursion for the out-of-plane
libration is increased. The full width at half-maximum of the remaining
spectral intensity in the difference spectra is 5 kHz, which corresponds
to an increase of approximately 4.3° in angular excursion between the
two temperatures.
Figure 3. (a) Components of the deuteron electric field gradient (Vxx,
Vyy, Vzz) and carbonyl chemical shift anisotropy (σ11, σ22, σ33) drawn
onto the molecular geometry of a trans amide. Both components Vyy
and σ22 are normal to the amide plane. Note that the motion observed
for several of the fast relaxing amides modulates the Vyy and Vzz
components as indicated by the ellipsoid. (b) The NH‚‚‚OC bond
length, the NH‚‚‚OC bond angle, and a Cartesian coordinate system
used to relate neighboring amides via Euler angles.
exhibit approximately static N-D deuterium line shapes, which
eliminates the possibility of large-amplitude molecular motions.
Temperature-dependent line shapes of the fast relaxing amides
indicate that the motion responsible for the relaxation is an out-
of-plane libration. Detailed studies from other labs have also
described an out-of-plane libration for the amide unit in
polybenzyl glutamate38 and nylon 66.29 Following the example
of the study of Usha et al.,38 we measured the line shape of a
short amide, lauryllactam, at 223 and 393 K (Figure 4). The
subtracted spectra (corresponding to the two temperatures) show
differences, especially at 87 kHz corresponding to the Vyy
direction of the electric field gradient at the deuteron. This
residual intensity can be related to the difference in the root
mean square of the polar angles, θ and φ, at different
temperatures.39 The reduction of the spectral intensity at the
y-component is consistent with an out-of-plane libration which
increases its angular excursion with an increase in temperature,
which can be described by δθ = sin-1(δω/(2ηe2qQ/h)). The
quadrupolar coupling constant of lauryllactam at 223 K is 195
kHz, its anisotropy is 0.17, and the fwhm of the residual spectral
density is 5 kHz; therefore the increase in angular excursion is
4.3° over 150 K, or 0.025 deg/K. This value is similar to that
reported for polybenzyl glutamate, which increased its angular
excursion linearly from 8° at 200 K to 12° at 400 K.38
Detailed studies of the thermal parameters from diffraction
data for glycine anhydride40 and urea41 indicate that the
amplitude of the out-of-plane libration can be as much as 8-12°,
similar to that estimated from deuterium line-shape analysis for
caprylolactam in our study, and to the values estimated for
PBG38 and nylon 66.29 Thus similar thermal factors in the
diffraction data are indicated for compounds with short and long
T1 deuterium values.
lolactam are shown in Figure 5: one with a delay near the T1
zero-crossing and the other with a dealy of 5 times the spin-
lattice relaxation time. The principal tensor elements corre-
sponding to the direction of the N-D bond vector are indicated
on the spectrum. It is clear that the z-component and the
y-component are fully relaxed at the zero crossing, whereas the
x-components are still inverted. Molecular motion produces
spectra density at the Larmor frequency and twice the Larmor
frequency of the deuteron (61 and 122 MHz at 9.4 T,
respectively) most efficiently when the amides are aligned with
their z and y tensor components along the applied field. T1 is
given by
1
T1
3 eqEQ 2
(ΩPL) )
[J1(Ω′,ω) + 4J2(Ω′,2ω)]
(1)
(
)
8
p
where Jm(ΩPL,ω), the spectral density at frequency ω for a
molecule at an orientation ΩPL with respect to the applied field,
is
J (Ω ,ω) ) ∞C (Ω′,t) cos(ωt) dt
(2)
∫
m
PL
m
0
The direction of motion can also be probed with anisotropic
relaxation measurements. Two T1 anisotropy spectra of capry-
and Cm(ΩPL,t) is the correlation function for the motion,

Spin-Lattice Relaxation Times in Secondary Amides
J. Phys. Chem. B, Vol. 102, No. 32, 1998 6255
Figure 6. Spin-lattice relaxation for methylene- and nitrogen-
deuterated caprylolactam as a function of temperature. The top graph
is simply T1 vs temperature for both samples where the circles (lower
curve) are for the R-methylene and the squares (upper curve) are for
N-D. The bottom graph is the inverse of the spin-lattice relaxation
time as a function of temperature squared where the circles (upper
curve) are for the R-methylene and the squares (lower curve) are for
N-D. The “slope” is proportional to the angular excursion, the
quadrupolar coupling constant, and the diffusion constant (see text for
further discussion). The errors for the temperature and rate are shown
as 1% and 15% RSD, respectively, although the error in most
measurements is much less.
Figure 5. Anisotropy spectra of deuterium-exchanged caprylolactam
collected with the relaxation delay set to the “zero-crossing” time and
at 8 times the spin-lattice relaxation time (spectra collected at room
temperature). Note that the wings at (145 kHz (marked Vzz) and the
shoulders at (86 kHz (marked Vyy) are fully relaxed at a 1.0 s delay,
whereas the horns at (63 kHz (marked Vxx) are still mostly inverted.
The rate of relaxation is determined by molecular motion at 1× and
2× Larmor frequency (60.8 and 121.6 MHz) and is dependent on the
polar angles θ and φ. The observed anisotropy in the relaxation rate
indicates that the Vyy and Vzz components are more effectively modulated
than the Vxx component. This observation is also consistent with an
out-of-plane libration which pivots about the Vxx component.
the electric field gradient, which would produce substantially
more modulation for the y- and z-components than for the
x-component. Anisotropic relaxation has been observed for
caprylolactam, for lauryllactam, for the cyclic monomer as well
as for caprolactam near its melting point (55 °C); the extent of
the anisotropy is quite modest for others which are considerably
less crystalline, such as polyalanine. In the more amorphous
materials it is possible that the model for the motion should be
closer to random motion in a cone. Simulation of the out-of-
plane rocking of the amide with a single correlation time does
not produce the amplitude of anisotropy observed in the spectra
for caprylolactam. To reproduce the spectra, we must use a
more complicated model, including motions on a wide range
of time scales (data forthcoming).
2
Cm(ΩPL,t) ) (F20)-2
F2nF2n′D (ΩCL)
am
(2)*
∑
nn′)-2
aa′)-2
Da(2′m) (ΩCL) D(2)*(ΩPC(0)) D(2)(ΩPC(t)) (3)
an
a′n
(2)*
am
D
(Ω) is the Wigner rotation matrix, Ω represents the set of
To clarify the nature of the motion, methylene units adjacent
to the amide for caprolactam and caprylolactam were specifically
deuterated (R-D lactams), and the corresponding deuterium T1
times were measured. In both cases the relaxation times for
the methylene-deuterated sample were shorter than the N-D
deuterated samples; for N-D caprolactam the T1 is 64 s, for
CD2-CO-NH-CD2 caprolactam the T1 is 45 s, for N-D
caprylolactam the T1 is 1.2 s, and for CD2-CO-NH-CD2
caprylolactam the T1 is 0.7 s (all at room temperature).29 Line
shapes for the methylene-deuterated compounds also resemble
static Pake patterns with QCC values of 170 kHz. Anisotropic
zero-crossing spectra of R-D caprylolactam exhibited Vzz
components that relax faster than the unresolved Vxx and Vyy
components. Carbon relaxation measurements showed similar
relaxation times for all methylenes, consistent with the proposal
that the entire molecule librates with a similar time scale and
angular extent.
Euler angles that transform one frame to another frame, and
13
x
F20 is the strength of the quadrupolar interaction ( 2/3eq).
The correlation function contains a time-dependent term that
describes the motion in a molecular fixed frame (the bracketed
term). The time-dependent Euler angles, ΩPC(t), describe the
motion in terms of the coordinate system of the molecule; these
angles rotate the principal axis system of the molecule to a
crystal fixed frame, and the crystal fixed frame is assumed to
be time independent. (For the principal axis system, the
direction from the N-D vector is the z-axis, and the vector in
the amide plane orthogonal to z is the x-axis.) The Euler angles,
Ω
CL, which are time independent (nonbracketed term), rotate
the molecular fixed frame into the laboratory frame. If the
motion is anisotropic such as an out-of-plane libration, then the
correlation function, the spectral density, and the relaxation will
be anisotropic as well and are explicit functions of the orientation
of the amide with respect to the applied field.
The temperature dependence of the deuterium T1 relaxation
rates for N-D and R-D caprylolactam was measured. In Figure
6, T1 values are plotted as a function of temperature. The T1
for both N-D and R-D decrease monotonically with temper-
Specific values of the anisotropic relaxation rates can be
calculated given a model of the molecular motion. For example,
we propose that the amide is executing an out-of-plane libration
about an axis approximately aligned with the x-component of

6256 J. Phys. Chem. B, Vol. 102, No. 32, 1998
Williams and McDermott
Figure 7. (left) Torsional angles, φ, ψ, and τ represented as arrows drawn onto hexamethylenedipropionamide and polyalanine. Note that in the
case of an R-helix the proton on the carbon alpha to the CO eclipses oxygen and the proton alpha to the NH is in a gauche conformation. In the
case of the hexamethylenedipropionamide the protons adjacent to both groups are in an all-trans conformation. The dihedral angles, φ ) C2′C1′-
NC(dO), C1′N-C(dO)C1, N(Od)C-C1C2, are defined in Figure 3. (right) Three views of the hexamethylenedipropionamide, two turns of polyalanine
(without protons or the side chain), and the cyclic monomer. The top row is a projection down the molecular axis defined in Figure 3. The middle
and bottom rows are projections down the y-axis and z-axis, respectively. For the cyclic monomer, the dihedral angles adjacent to the amide unit
must take on a gauche conformation in order to complete the ring. For polyalanine in the R-helix conformation the dihedrals after the amide unit
are also gauche in order to satisfy hydrogen bonding with the i+4 and i-4 residue. For hexamethylenedipropionamide, there are no internal
constraints and the methylene chain takes on an all-trans conformation.
ature. The relaxation times increase more slowly with decreas-
ing temperature for the CD group compared to the N-D
deuteron.
the CD2 groups of the alkane chain undergo a small angle
“wobble” of approximately 20-30° with the correlation time
of 10-10-10-11 s.42 The deuterium relaxation data of cyclo-
pentane-d10 also indicate a small 5-10° libration that is
quenched on passing from the phase II (a glassy state) to phase
III (a crystalline state).43 A small-angle libration of the
tryptophan side chain in bacteriorhodopsin44 and in triose
phosphate isomerase45 has been invoked to explain the fast
relaxation rate.
Spectral Density and Librational Models. In this section
we discuss models relating the relaxation times to characteristic
time constants for the libration. The models are derived largely
from the literature. At the outset we were surprised to observe
such short relaxation times associated with a librational motion.
Typically, motions with large amplitudes with near equal
populations between the sites and rates near the Larmor
frequency (60.8 MHz) have fast relaxation rates (e.g., the methyl
group three-site hop or a trans-gauche isomerization in lipid
bilayers). An out-of-plane libration, as suggested by our data,
is generally assumed to be of low amplitude and in the low
IR/Raman frequency bands (10-50 cm-1 or 47-238 GHz).
Thus, one might have expected all deuterated amides to have
long deuterium T1 relaxation times. Nonetheless, small angular
excursions have been proposed previously to explain small T1
values in other studies. Presumably, the reason that librations
are able to act as a relaxation mechanism is that they become
overdamped due to structural and conformational restraints. In
this study, we are use our structural survey together with
previously published models to understand the molecular origin
of this damping.
The assumption that the principal contribution to the spectral
density is on a time scale very fast compared to the Larmor
frequency was confirmed by measuring the anisotropic relax-
ation of quadrupolar order.56 For a fast-limit motion the
relaxation times should follow the relation T1Q/T1Z ) 5/3; the
rates were approximately in this ratio for the fast-relaxing amide
caprylolactam HCl, and others are currently under investigation.
Theoretical treatments of the out-of-plane libration have been
presented previously.31,38 A model based on the stochastic
diffusion equation has been shown to be essentially equivalent
to a strongly damped Langevin oscillator with a harmonic
driving force, where the motion is assumed to be on a fast time
scale and involve small angle fluctuations; both models appear
to explain the spin-lattice relaxation behavior.31,38,42,46 A model
based on the stochastic diffusion equation predicts a correlation
time τ(Ω) ≈ 12 θ2Lib(Ω) /π2D(Ω) (eq 5) and an aniso-
tropic (frequency-independent) spectral density Jm(Ω) )
There are several previous examples for librational motions
providing relaxation mechanisms. For example, a detailed
relaxation study of n-nonadecane-d40/urea clathrate indicates that

Spin-Lattice Relaxation Times in Secondary Amides
J. Phys. Chem. B, Vol. 102, No. 32, 1998 6257
3/2 θ2Lib τ(Ω) (eq 6)46 where D(Ω) is the (anisotropic) diffusion
constant, the tensorial angular excursion θLib(Ω) describes an
oscillation which is primarily about the x-axis in the molecular
frame, and Ω represents the Euler angles which describe the
orientation converting the principle axis system to the lab frame
for each crystallite and must be used to compute the anisotropic
parameters Θ, D, and τ. An alternative model is based on a
driven harmonic oscillator subject to friction.38,47,48 The libra-
tion of the amide plane occurs along a coordinate, η, which
is the difference of the Ramachadran angles, η ) (φ -
θ2(0) T (eq 10), the temperature dependence of T1 is given by
η
1
15 eEqQ 2
2T2
D
=
θ2η(0)
(11)
(
)
16
p
T1(Ω′)
From eq 11, the inverse of T1 vs the square of temperature
should produce a straight line with a slope equal to quotient of
the angular excursion and the diffusion constant. In Figure 6,
we show that such a plot for the amide-deuterated sample is
relatively linear, but the R-labeled compound shows some
deviation. The change in libration angle with temperature for
the amide-deuterated sample was found from the width in
residual density in the line shape collected at two temperatures
and is comparable to that found in the study of Usha et al.38
Using the value for θ2(0) from our data, 1.5 × 10-4 rad2/K
(1.1 × 10-4 rad2/K38), and the quadrupole coupling constant of
200 kHz, the slope obtained from the fit gives 1.4 × 10-5 for
the amide-deuterated sample giving a diffusion constant of 2.9
× 108 s-1. The slope for the R-deuterated sample is 5.4 ×
10-5, although it appears to deviate from linearity. This faster
relaxation rate and slope for the R-deuterated compound is
consistent with a smaller moment of interia, which predicts a
greater amplitude in the angular excursion or a stronger frictional
force or both. Careful line-shape analysis of nylon 66 from
previous work of English et al. indicates that a larger angular
excursion for the methylenes is most likely to be responsible
for this difference between the amide and the flanking meth-
ylenes.29
38,49
x
æ) 2/2.
If this motion is subject to strong frictional
damping with a damping constant λ, and the angular fluc-
tuation is assumed to be small, it has been shown that the
spectral density is Jm ) 3 θ2η (λ/ωη2) 38 (eq 7), where the damp-
ing constant λ ) ú/2Iη, the “undamped” frequency ωη
)
(κT/Iη θ2η )1/2 (eq 8), Iη is the interia, κ is the Boltzmann
constant, T is the temperature, and ú is the friction coefficient.47
Here the anisotropic spectral density and relaxation rates result
from the fact that the normal mode must be projected onto the
molecular axis system which in turn relates to the laboratory
frame in a different way for each crystallite. The final
expression for the spectral density for the Langevin oscillator
model in the limit of strong friction can be compared with the
one that results from the stochastic diffusion model using the
Einstein relation, D ) kbT/ú (eq 9); the expressions differ then
only by a small scaling constant. This similarity is interesting
since the physical assumptions are very different; the spectral
density of the damped Langevin oscillator would be centered
about the oscillator frequency except that it extends to lower
frequencies due to a frictional force, while the spectral density
for the stochastic diffusion is inherently centered around zero
frequency.38,48 In the Langevin oscillator model, amides with
long relaxation times would be described by weak frictional
coefficients, so that the oscillator approaches the inertial limit
and the spectral density is strongly peaked at the oscillator
frequency.48 Amides with short relaxation times would be
described by strong frictional coefficients with “white” spectral
density, which is analogous to the result derived from the
stochastic diffusion model. In summary, we used previously
derived theoretical models to recast our spin-lattice relaxation
data in terms of a picture in which amides with certain
conformations are usually subject to strong local frictional forces
and librate diffusively and amides with other conformations are
subject to weak local friction and librate harmonically, with rates
far above the NMR time scale.
Dihedral Angles and Spin-Lattice Relaxation Rates. In
summary, our data suggest that some but not all amides in
crystalline environments show a slow librational process on the
time scale which leads to spin-lattice relaxation for the amidic
deuteron. All amides that exhibit relaxation have flanking alkyl
groups (larger than a methyl group), and most have the flanking
dihedral values near -60, so that gauche interactions are
expected.
Assuming that the angular excursion is similar for amides
with different spin-lattice relaxation rates, which is supported
by crystallographic thermal factors, the difference in these spin-
lattice relaxation rates is due to the difference in friction
coefficient or diffusion constants. These constants can be
defined in molecular terms using the fluctuation-dissipation
theorem, which states that the diffusion constant is related to
the correlation function of the random fluctuating field. Dif-
fusion constants for solute molecules have been calculated; the
random fluctuating field is usually considered in terms of
random collisions of a solute molecule with the solvent
molecules, and the diffusion constant is related to the autocor-
relation function of the particle’s velocity.53-55 In a crystalline
solid, these collisions would not be spatially random. The
fluctuating field experienced by the amidic deuteron is related
to its neighbor’s motion, the adjacent methylene groups, and
crystallographic neighbors. If the random fluctuating field is
harmonic and at a frequency very different than the frequency
of the amide libration, its spectral density will not be effective
in damping. If the random fluctuating field is stochastic or
strongly overdamped, it could have significant overlap with the
librational frequency resulting in a strong frictional force or
small diffusion constant. The difference between the fast
relaxing and the slow relaxing amides is proposed to be the
extent of coupling to modes in flanking regions. It is striking
that flanking dihedral groups are needed for the motion to appear
on the NMR time scale. We observed a correlation between
flanking dihedral angles and relaxation rates of the amidic
With this model and the estimate of the libration angle we
can obtain a time constant (or a damping constant) for the
libration from our relaxation data. Previous work describing
the out-of-plane libration of the amide reported typical values
for the correlation time of approximately τc ) 8 ps. With the
assumption that the angle of libration is approximately 10°, we
also obtain a correlation time of the same order of magnitude
(approximately 14 ps, 60 GHz or 2 cm-1) for a T1 of 1 s. For
relaxation times of 10 ms to 500 s the correlation times range
from 1.4 ns to 140 fs (600 MHz to 6 THz, or 0.2 to 200 cm-1
.
For comparison, computational studies of proteins using mo-
lecular dynamics or normal mode methods have found a number
of modes with frequencies in the 1-100 cm-1 range.10,11,51,52
Many of the modes that involve rotations or distortions of the
amide have frequencies around or below 50 cm-1
.
Since our temperature-dependent data clearly indicate that
the motion is not an activated process but a librational process,
it is worthwhile to discuss how T1 is related to temperature.
From the equations above and rewriting eq 8 as θ2η(T) )

6258 J. Phys. Chem. B, Vol. 102, No. 32, 1998
Williams and McDermott
spin-lattice relaxation time
cross polarization-magic angle spinning
chemical shift anisotropy
hydrogen bond
T1
deuteron for a wide range of amides. For the fast relaxing
deuterons, with φ and ψ values around -60°, the R-methylene
protons eclipse amidic oxygen and are gauche to the amidic
proton (Figure 7). The groups adjacent to the R-methylene
groups are also gauche to their neighbors. We conclude that
the flanking dihedral angles in secondary amides somehow
influence the damping of the out-of-plane libration of the amide
plane, possibly involving nonbonded interactions such as the
eclipsing and gauche.
In a normal mode dynamics simulation of bovine pancreatic
trypsin inhibitor, the displacements of φ and ψ were correlated
against each other and against the dihedrals of adjacent amides.51
Significant correlations for R-helix secondary structure were
observed for dihedrals four units away in either direction, but
notably absent for the â-sheet secondary structure. The cor-
relation value of the φ/ψ angle of the central amide unit, ∆φ0
∆ψ0 , was -0.30 for an R-helix and 0.0 for a â-sheet.
On the basis of our observations one might expect damping
and modification of amide librational dynamics to encapsulate
the slower time scales (nanosecond) for R-helical structures;
this damping might be important for facilitating an even slower,
larger amplitude and more functionally relevant motions. In
contrast, â-structures are predicted to exhibit stiffer and more
harmonic motions which remain on a time scale far separated
from that of functionally relevant processes. This predicted
difference in the time scales of backbone motions should be
tested with SSNMR studies on selectively labeled proteins.
CPMAS
CSA
H-bond
HPMDB
heptamethylenedibenzamide
hexamethylenedibenzamide
hexamethylenediacetamide
hexamethylenedipropionamide
1,6-diazacyclododecane-7,12-dione
1,8-diazacyclotetradecane-2,7-dione
2-azacyclononanone
HXMDB
HXMDA
HXMDP
polyamide-4,6
cyclic monomer
caprylolactam
caprolactam
lauryllactam
2-azacycloheptanone
2-azacyclotridecanone
homoazaadamantone 4-azatricyclo[4.3.1.1]undecan-5-one
nylon-4,4
nylon-6,6
tetramethylenedibutylamide
hexamethylenedihexamide
References and Notes
(1) Englander, S. W.; Englander, J. J.; McKinnie, R. E.; Ackers, G.
K.; Turner, G. J.; Westrick, J. A.; Gill, S. J. Science 1992, 256, 1684-
1687.
(2) Kossiakoff, A. A. Nature 1982, 296, 713-721.
(3) Austin, R. H.; Beeson, K. W.; Eisenstein, L.; Frauenfelder, H.;
Gunsalus, I. C. Biochemistry 1975, 14, 5355-5373.
(4) Stezowski, J. J.; Peachey, N. M.; Goebel, P.; Eckhardt, C. J. J.
Am. Chem. Soc. 1993, 115, 6499-6505.
(5) Frauenfelder, H.; Sligar, S. G.; Wolynes, P. G. Science 1991, 254,
1598-1603.
(6) Dlott, D. D. Optical Phonon Dynamics in Molecular Crystals;
Annual Reviews: 1986.
Conclusions
We have measured librational dynamics of amide groups in
crystalline solids and compared these motions with a variety of
structural and electronic properties. We observed fast deuterium
spin-lattice relaxation rates associated with a damped or slow
amide libration when the amide is alkyl substituted on both sides
and furthermore if both dihedral angles adjacent to the amide,
φ and ψ, were approximately -60. We also found that 13C
CSA, deuterium quadrupolar coupling constants, and CO and
NH stretch frequencies are not particularly sensitive to the
hydrogen bond geometry or modest changes in length. We
postulate that the difference in deuterium spin-lattice relaxation
rates for the various amides is related to the anharmonicity of
the amide deflection and consequent damping of the motion,
resulting in significant spectral density on the NMR time scale.
The damping is apparently unrelated to hydrogen bonding but
strongly related to flanking alkyl groups.
(7) Fanconi, B.; Small, E. W.; Peticolas, W. L. Biopolymers 1971, 10,
1277-1298.
(8) Bee, M. Quasielastic Neutron Scattering; 1st ed.; Adam Hilger:
Bristol, 1988.
(9) Jonscher, A. K. J. Mater. Sci. 1981, 2037-2060.
(10) Pleiss, J.; Jahnig, F. Biophys. J. 1991, 59, 795-804.
(11) Perahia, D.; Levy, R. M.; Karplus, M. Biopolymers 1990, 29, 645-
677.
(12) Spiess, H. W. NMR: Basic Princ. Prog. 1980, 15, 59-213.
(13) Vold, R. R.; Vold, R. G. AdV. Magn. Opt. Resn. 1991, 16, 85-
171.
(14) Griffin, R. G. Methods of Enzymology; Academic Press: New York,
1981; Vol. 72, pp 108-174.
(15) Rabbani, S. R.; Edmonds, D. T.; Gosling, P. J. Magn. Reson. 1987,
72, 422-433.
(16) Brown, T. L.; Butler, L. G.; Curtin, D. Y.; Hiyama, Y.; Paul, I. C.;
Wilson, R. B. J. Am. Chem. Soc. 1982, 104, 1172-1177.
(17) Bersohn, R. J. Chem. Phys. 1960, 32, 85-88.
(18) Hunt, M. J.; Mackay, A. L. J. Magn. Reson. 1976, 22, 295-301.
(19) Marks, B. S.; Schweiker, G. C. J. Polym. Sci. 1960, 43, 229-239.
(20) Gaymann, R. J.; Harkema, S. J. Polym. Sci. Phys. Ed. 1977, 15,
587-590.
(21) Vulic’, I.; Hooft, R. W. W.; Kroon, J. J. Crystallogr. Spectrosc.
Res. 1993, 23, 917-920.
(22) Herzfeld, J.; Berger, A. J. Chem. Phys. 1980, 73, 6021-6030.
(23) The Role of Hydrogen Bonding in Molecular Assemblies; Bernstein,
J., Etter, M. C., Leiserowitz, L., Eds.; VCH: New York, 1994.
(24) Leiserowitz, L.; Tuval, M. Acta Crystallogr. 1978, B34, 1230-
1247.
Acknowledgment. We gratefully acknowledge Sharon
Rozovsky for quadrupolar order measurements and for carefully
reading the manuscript. We also thank Dr. Titan Gu for advice
on CSA simulations. We thank Professor Richard Wittebort at
the University of Louisville, Kentucky, for conversations about
the dynamics of the amide unit; Professor Ged Parkin, Columbia
University, for collecting X-ray data and solving the N-tert-
butylpropionamide crystal structure; Dr. Ivan Vulic’ at DSM,
The Netherlands, for providing a sample of the polyamide-4,6
model compound; and Dr. Nancy Triggs, Notre Dame, and
Professor James Valentini, Columbia University, for discussion
of amide hydrogen bonding.
(25) Berkovitch-Yellin, Z.; Leiserowitz, L. J. Am. Chem. Soc. 1980, 102,
7677-7690.
(26) Dauber, P.; Hagler, A. T. Acc. Chem. Res. 1980, 13, 105-112.
(27) Taylor, R.; Kennard, O.; Versichel, W. Acta Crystallogr. 1984, B40,
280-288.
(28) Benedetti, E.; Ciajolo, M. R.; Corradini, P. Eur. Polym. J. 1973, 9,
101-109.
(29) Hirschinger, J.; Miura, H.; Gardner, K. H.; English, A. D.
Macromolecules 1990, 23, 2153-2169.
Supporting Information Available: Spin-lattice relaxation
rates and CSA tensor elements (4 pages). Ordering information
is given on any current masthead page.
(30) Dunitz, J. D.; Winkler, F. K. Acta Crystallogr. 1975, B31, 251-
263.
(31) Ando, S.; Ando, I.; Shuji, A.; Ozaki, T. J. Am. Chem. Soc. 1988,
110, 3380-3386.
(32) Asakawa, N.; Kuroki, S.; Kurosu, H.; Ando, I.; Shoji, A.; Ozaki,
T. J. Am. Chem. Soc. 1992, 114, 3261-3265.
Abbreviations
SSNMR
solid state NMR

Spin-Lattice Relaxation Times in Secondary Amides
J. Phys. Chem. B, Vol. 102, No. 32, 1998 6259
(33) Gu, Z.; Zambrano, R.; McDermott, A. J. Am. Chem. Soc. 1994,
116, 6368-6372.
(44) Kinsey, R. A.; Kintanar, A.; Oldfield, E. J. Biol. Chem. 1981, 256,
9028-9036.
(34) Llinas, M.; Wilson, D. M.; Klein, M. P. J. Am. Chem. Soc. 1977,
99, 6846-6850.
(45) Williams, J. C.; McDermott, A. E. Biochemistry 1995, 34, 8309-
8319.
(35) Lumsden, M. D.; Wasylishen, R. E.; Eichele, K.; Schindler, M.;
Penner, G. H.; Power, W. P.; Curtis, R. D. J. Am. Chem. Soc. 1994, 116,
1403-1413.
(46) Wittebort, R. J.; Szabo, A. J. Chem. Phys. 1978, 69, 1722-1736.
(47) Chandrasekhar, S. ReV. Mod. Phys. 1943, 15, 1-89.
(48) Szabo, A. J. Chem. Phys. 1984, 81, 150-167.
(49) Peticolas, W. L.; Kurtz, B. Biopolymers 1980, 19, 1153-1166.
(50) Spiess, H. J. Chem. Phys. 1980, 72, 6755-6762.
(51) Levitt, M.; Sander, C.; Stern, P. S. J. Mol. Biol. 1985, 181, 423-
447.
(36) Winkler, F. K.; Dunitz, J. D. Acta Crystallogr. 1975, B31, 268-
269.
(37) Chiba, T. Bull. Chem. Soc. Jpn. 1965, 38, 259-263.
(38) Usha, M. G.; Peticolas, W. L.; Wittebort, R. J. Biochemistry 1991,
30, 3955-3962.
(52) Brooks, B.; Karplus, M. Proc. Natl. Acad. Sci. U.S.A. 1983, 80,
6571-6575.
(39) Henry, E. R.; Szabo, A. J. Chem. Phys. 1985, 82, 4753-4761.
(40) Lonsdale, K. Acta Crystallogr. 1961, 14, 37-42.
(41) Swaminathan, S.; Craven, B. M.; McMullan, R. K. Acta Crystallogr.
1984, B40, 300-306.
(53) Straub, J. E.; Borkovec, M.; Berne, B. J. J. Phys. Chem. 1987, 91,
4995-4998.
(54) Berne, B. J.; Borkovec, M.; Straub, J. E. J. Phys. Chem. 1988, 92,
3711-3725.
(42) Greenfield, M.; Vold, R. L.; Vold, R. R. Mol. Phys. 1989, 66, 269-
298.
(43) Mack, J. W.; Torchia, D. A. J. Phys. Chem. 1991, 95, 4207-4213.
(55) Zhang, Y.; Bull, T. E. Phys. ReV. E 1994, 49, 4886-4902.
(56) Williams, J. C. Thesis, Columbia University, 1995.