Vibrational dynamics of N-H, C-D, and C=O modes in formamide

Article abstract of DOI:10.1063/1.2162165

By means of heterodyned two-dimensional IR photon echo experiments on liquid formamide and isotopomers the vibrational frequency dynamics of the N-H stretch mode, the C-D mode, and the C=O mode were obtained. In each case the vibrational frequency correlation function is fitted to three exponentials representing ultrafast (few femtoseconds), intermediate (hundreds of femtoseconds), and slow (many picoseconds) correlation times. In the case of N-H there is a significant underdamped contribution to the correlation decay that was not seen in previous experiments and is attributed to hydrogen-bond librational modes. This underdamped motion is not seen in the C-D or C=O correlation functions. The motions probed by the C-D bond are generally faster than those seen by N-H and C=O, indicating that the environment of C-D interchanges more rapidly, consistent with a weaker C-D...O=C bond. The correlation decays of N-H and C=O are similar, consistent with both being involved in strong H bonding.

Full text of DOI:10.1063/1.2162165

Vibrational dynamics of N–H, C–D, and CO modes in formamide
Jeong-Hyon Ha, Yung Sam Kim, and Robin M. Hochstrasser
Citation: J. Chem. Phys. 124, 064508 (2006); doi: 10.1063/1.2162165
View online: http://dx.doi.org/10.1063/1.2162165
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THE JOURNAL OF CHEMICAL PHYSICS 124, 064508 ͑2006͒
Vibrational dynamics of N–H, C–D, and CBO modes in formamide
Jeong-Hyon Ha, Yung Sam Kim, and Robin M. Hochstrassera͒
Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104
͑Received 2 November 2005; accepted 5 December 2005; published online 14 February 2006͒
By means of heterodyned two-dimensional IR photon echo experiments on liquid formamide and
isotopomers the vibrational frequency dynamics of the N–H stretch mode, the C–D mode, and the
CvO mode were obtained. In each case the vibrational frequency correlation function is fitted to
three exponentials representing ultrafast ͑few femtoseconds͒, intermediate ͑hundreds of
femtoseconds͒, and slow ͑many picoseconds͒ correlation times. In the case of N–H there is a
significant underdamped contribution to the correlation decay that was not seen in previous
experiments and is attributed to hydrogen-bond librational modes. This underdamped motion is not
seen in the C–D or CvO correlation functions. The motions probed by the C–D bond are generally
faster than those seen by N–H and CvO, indicating that the environment of C–D interchanges
more rapidly, consistent with a weaker C–D¯OvC bond. The correlation decays of N–H and
CvO are similar, consistent with both being involved in strong H bonding. © 2006 American
Institute of Physics. ͓DOI: 10.1063/1.2162165͔
I. INTRODUCTION
formamide. Excitation of the N–H mode induces relaxation
into low-frequency modes of the liquid.10 This relaxation
causes distinctive signals in 2D IR photon echo experiments
that permit measurement of the correlation of the frequency
distributions of the N–H mode and the modes to which it
transfers energy. These modes were found to be strongly cor-
related. Since the N–H mode distribution width is largely
determined by H-bonding strength, with weak H bonds giv-
ing higher-frequency transitions, the relaxed modes were
presumed to have the same characteristic. Librations and
stretching of H-bonded molecules were reasonable candi-
dates to account for the observed strong correlation. How-
ever, the fastest component of frequency correlation function
was not fully explored in these experiments. Therefore, we
have conducted a more extensive study of this short-time
regime. The C–H or C–D mode is expected to be involved in
the hydrogen-bonding network and it was thought that its
dynamics might provide a more detailed picture of the liquid
formamide structure. To the best of our knowledge, there are
no previous reports of the ultrafast nonlinear IR photon echo
Interactions and angular relationships between vibrators
and the dynamics of their frequency fluctuations can be mea-
sured by multidimensional IR spectroscopy.1–4 The fre-
quency bandwidth of the excitation infrared pulses is con-
trolled to span the distribution of interacting states in these
experiments. The heterodyned two-dimensional ͑2D͒ IR
spectroscopy has now been applied to the measurement of
structures and dynamics of various peptides and proteins.5–7
These systems have repeating units giving rise to sets of
nearly degenerate vibrations. Similar situations in solids, liq-
uids, or glasses are encountered where the vibrational states
of different molecules couple together to form excitation
bands. Therefore, the heterodyned 2D IR spectroscopy also
can help visualize the structure and dynamics of liquids.8–12
Although the conventional Fourier transform infrared
͑FTIR͒, stationary and time-resolved nonlinear Raman meth-
ods, and other forms of ultrafast vibrational spectroscopy
have had many applications to liquids and have contributed
substantially to the understanding of liquid state dynamics,
the method of 2D IR heterodyned echo spectroscopy used in
this paper is an interesting complement to these
approaches.13 Here, we discuss experiments using the ul-
trafast heterodyned 2D IR to examine the H-bonded liquid,
formamide.
of
a C–D mode. The C–D transition lies in the
1800–2600 cm−1 region where there are few interferences
from other modes even in proteins.20–23 Through the C–D
mode, the role of the CD¯O hydrogen bonding in the over-
all structure and dynamics of formamide can be examined.
This type of hydrogen bond is a weak attractive interaction
whose existence has been experimentally predicted and dem-
onstrated both by spectroscopic and crystallographic
techniques.24–28 Recently, the possible importance of the
CH¯O hydrogen bonding in the dynamics and structures of
membrane proteins has been recognized.29,30 The amide-I
͑carbonyl stretch͒ is one of the major IR active modes in
formamide liquid structure.31,32 This mode is involved in the
extended hydrogen-bonding network and, therefore, its be-
havior is expected to be highly correlated with other modes
in liquid formamide.
Formamide is the smallest unit in a peptide chain and
forms extended hydrogen-bond networks in its liquid
state.14–18 The intramolecular hydrogen bonds formed by
peptides determine the secondary structures of protein; so
from this standpoint liquid formamide is particularly
interesting.19 It has four different types of chemical bonds
whose dynamical properties can be accessed by nonlinear
infrared studies of its IR active modes. In a previous experi-
ment, we probed the dynamics and motions of N–H bond in
a͒
Electronic mail: hochstra@sas.upenn.edu
0021-9606/2006/124͑6͒/064508/10/$23.00
124, 064508-1
© 2006 American Institute of Physics
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064508-2
Ha, Kim, and Hochstrasser
J. Chem. Phys. 124, 064508 ͑2006͒
˜
The present work examines the equilibrium dynamics of
the N–H stretching, C–D stretching, and CvO stretching
motions of liquid formamide. Their frequency correlation
functions were compared in order to find an improved ex-
perimental description of the immediate environment to
which each of these modes is coupled in the liquid.
mation of S͑␶,T,␻ ͒ on the ␶ axis generated the complex 2D
t
IR spectra, S>͑␻ ,T,␻ ͒. The real parts of the projections can
␶
t
be obtained by taking discrete sums over the ␻ data sets
␶
͓͚ S>R͑␻ ,T,␻ ͒ei␾ and ͚ S>NR͑␻ ,T,␻ ͒ei␾ ͔. These pro-
R
NR
␻
␶
t
␻
␶
t
␶
␶
jections reproduce the measured pump/probe spectrum at T,
˜
defined as S͑␶=0,T,␻ ͒ by adjusting the phases ␾ and ␾
t
R
NR
to find the best agreement. With the proper phases included
the sum of the rephasing and nonrephasing signals generates
a pure 2D IR absorptive spectrum, or more simply 2D IR
II. EXPERIMENT
A. Sample preparation
spectrum. A profile along ␶ at each detection frequency ␻
can be obtained from each of the multidimensional data sets,
t
1. DCONHD in deuterated formamide „DCOND2…
The sample ͑2% DCONHD͒ was made by fully mixing
the formamide-H3 with liquid deuterated formamide-D3.
Both reagents were from Aldrich. The FTIR absorption spec-
tra were obtained at 1 cm−1 resolution. The sample optical
density for the echo experiments at the peak was ϳ0.3 in a
CaF2 cell of path length of 12 ␮m.
˜
such as ͉S͑␶,T,␻ ͉͒. This function gives a complete set of
t
T-dependent data at each ␻t, which is similar to that obtained
from the stimulated three-pulse photon echo signal. We char-
acterize each profile by the value of ␶ at which it has a
maximum: the so-called peak shift.
2. 5–6 ␮m region experiment
2. DCONH2 in formamide „HCONH2…
Fourier-transform-limited 80 fs pulses with center fre-
quencies 2169 and 1685 cm−1 were used in the 2D IR ex-
periments. Three 400 nJ IR pulses with wave vectors k1, k2,
and k3 were incident on the sample, and the phase-matched
signal at wave vector −k1+k2+k3 was detected by heterodyn-
ing it with a local oscillator pulse that always preceded the
signal pulse by a fixed interval of ϳ1.0 ps. The signal and
local oscillator pulses were combined at the focal plane of a
monochromator having a 64-element MCT array detector ͑IR
Associates, Inc.͒. Each detector element is 200 ␮m in width
and 1 mm in height. The focal length of the monochromator
is 270 mm and the groove density of the grating used in our
experiments was 50 lines/mm. The raw data collected using
this method were in the form of a two-dimensional array of
time ␶, in 2 fs steps from −2.0 to 2.0 ps, and wavelength in
ϳ6 nm steps. The procedures to obtain 2D IR absorptive
spectra and peak shift profile were as described above.
In all experiments there were nonresonant background
signals that could be seen when the pulse center frequency
was not centered on a transition. However, the resonant sig-
nal completely dominated the nonresonant signal in all cases.
This sample ͑8% DCONH2͒ was prepared by mixing the
liquids formamide-D3 and formamide-H3. The optical den-
sity in the echo experiment was ϳ0.15 in a path length of
25 ␮m. The integrated absorption cross section of the C–D
mode was 16% larger for this sample than formamide-D3 in
CHCl3.
3. HCONH2 in bulk formamide
A thin film of pure HCONH2 having an optical density
of ϳ0.1 at the carbonyl peak was prepared by spreading the
liquid formamide between CaF2 windows.
B. Laser instrument and signal definitions
1. 3 ␮m region experiment
The experiments were performed with the same instru-
ments and methods reported previously.10 Briefly, the pulses
in the 3 ␮m region were generated by a Ti:sapphire regen-
erative amplifier ͑Spectra Physics Spitfire͒ and a homemade
IR optical parametric amplifier ͑OPA͒.33 The resulting 3.5 ␮J
pulses of duration ϳ75 fs had a full spectral width at half
maximum of 190 cm−1 and were centered at 3.3 ␮m
͑3335 cm−1͒. The IR pulse was divided into three excitation
beams ͑k1,k2,k3͒ that generated the IR echo. A weak local
oscillator beam ͑kLO͒ heterodyned the generated echo field
and was sent to a monochromator having a liquid-N2-cooled
32-element mercury cadmium telluride ͑MCT͒ array detector
at the focal plane. The local oscillator beam and echo signals
were separated by 500 fs to facilitate the spectral resolution
and data processing. The heterodyned IR photon echo signal
III. RESULTS
A. DCONHD in deuterated formamide „DCOND2…
The linear IR spectrum ͑solid line͒ of 2% DCONHD in
deuterated formamide, DCOND2, in the N–H stretch vibra-
tion region is shown in Fig. 1. The broad featureless back-
ground coming from fully deuterated formamide is removed
from the measured linear IR-absorption spectrum. The N–H
band position is shifted to a frequency that indicates that the
N–H groups are hydrogen bonded. The IR pulses have suf-
ficient bandwidth to span the fundamental 0→1 transition as
shown in Fig. 1.
was measured as a function of the frequency ␻ detected at
t
32 discrete wavelengths, the time interval between the first
and second pulses, ␶, and the time interval between second
and third pulses, T. The echoes directed into −k1+k2+k3 with
k1 arriving at the sample before/after k2 are referred to as
rephasing/nonrephasing types for which we use the symbols
R/NR. We obtained the three-time signal S͑␶,T,t͒ by nu-
merical Fourier transformation on the monochromator fre-
quency axis. The inverse Fourier transformation on the t axis
From the heterodyned 2D IR signals, we can obtain the
˜
magnitude of ͉S͑␶,T,␻ ͉͒ by Fourier transforming the three-
t
time photon echo signal along the t axis. The profile of the
˜
quantity ͉S͑␶,T,␻ ͉͒ vs ␶ is peaked at finite values of ␶ for
t
each value of T. The peak shifts in Fig. 2 were obtained from
−1
˜
˜
of S͑␶,T,t͒ defines S͑␶,T,␻ ͒. A numerical Fourier transfor-
the sums of ͉S͑␶,T,␻ ͉͒ over ␻ from 3277 to 3357 cm for
t
t
t
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064508-3
Vibrational dynamics in formamide
J. Chem. Phys. 124, 064508 ͑2006͒
ways are included that involve the excitation ͑␯=0→␯=1͒
of N–H modes when the system is in the ͉y͘ states. In the
simulations, the response functions can be written in terms of
the four relaxation functions G:
GRxx͑␶,T,t͒ = − gxx͑␶͒ + gxx͑T͒ − gxx͑t͒ − gxx͑T + ␶͒
− gxx͑t + T͒ + gxx͑␶ + T + t͒,
͑1͒
͑2͒
͑3͒
͑4͒
GRxy͑␶,T,t͒ = − gxx͑␶͒ + gxy͑T͒ − gyy͑t͒ − gxy͑T + ␶͒
− gxy͑t + T͒ + gxy͑␶ + T + t͒,
NR
G
͑␶,T,t͒ = − gxx͑␶͒ − gxx͑T͒ − gxx͑t͒ + gxx͑T + ␶͒
+ gxx͑t + T͒ − gxx͑␶ + T + t͒,
xx
NR
xy
G
͑␶,T,t͒ = − gxx͑␶͒ − gxy͑T͒ − gyy͑t͒ + gxy͑T + ␶͒
+ gxy͑t + T͒ − gxy͑␶ + T + t͒.
FIG. 1. Linear IR-absorption spectrum ͑solid line͒ of the N–H stretch vibra-
tion of DCONHD in formamide-D3. The dotted line shows the spectrum
obtained by Fourier transformation of the response function ͓Eq. ͑17͔͒. The
dash-dot line indicates the laser-pulse spectrum.
The simplest forms for the relevant responses R1 through R8,
neglecting coherence transfer, involve the fundamental fre-
quencies ␻10, the anharmonicity ⌬, the foregoing G factors,
and the conditional probability factors, as follows:
R
˜
each ␶ value. The center frequency of the signal ͉S͑␶,T,␻ ͉͒
t
along ␻ is at 3317 cm−1, which is at the peak of the funda-
t
R1 = P͑1,0͉1,T͒e−i␻
eG
,
͑5͒
͑6͒
͑7͒
͑8͒
͑t−␶͒
͑␶,T,t͒
10
xx
mental transition. The peak shift decays after T=0 and it
exhibits a damped oscillation that increased the peak shift to
a maximum at T=160 fs after which it decayed, as shown in
Fig. 2.
R
͑t−␶͒
͑␶,T,t͒
R2 = P͑0,0͉0,T͒e−i␻
eG
,
10
xx
R
We simulated the linear spectrum and the peak shift pro-
file with the methods reported previously,10 and summarized
again here, in which the photon echo response functions con-
sist of a set of eight Liouville pathways. Four of these path-
ways describe the rephasing and the other four describe the
nonrephasing contributions. The excited-state and ground-
state population gratings contribute four pathways. Included
also are the two pathways involving ␯=1 to ␯=2 transitions.
The T1 relaxation of the ␯=1 excitation of N–H populates
other unknown vibrational states of the system,10 which we
symbolized by ͉y͘. We showed that these ͉y͘ states probably
involve relative motion of formamide molecules. Two path-
−i͑͑␻ −⌬͒t−␻ ␶͒
͑␶,T,t͒
R3 = − 2P͑1,0͉1,T͒e−␣t/T
e
eG
,
1
10
10
xx
R
−⌬ ͒t−␻ ␶͒
͑␶,T,t͒
R = −
P͑1,0͉y,T͒e−i͑͑␻
eG
,
10
y
10
xy
͚
4
y
NR
͑t+␶͒
͑t+␶͒
͑␶,T,t͒
͑␶,T,t͒
R5 = P͑1,0͉1,T͒e−i␻
R6 = P͑0,0͉0,T͒e−i␻
eG
eG
,
,
͑9͒
͑10͒
͑11͒
͑12͒
10
10
xx
NR
xx
NR
−i͑͑␻ −⌬͒t+␻ ␶͒
͑␶,T,t͒
R7 = − 2P͑1,0͉1,T͒e−␣t/T
e
eG
,
1
10
10
xx
NR
xy
−⌬ ͒t+␻ ␶͒
͑␶,T,t͒
P͑1,0͉y,T͒e−i͑͑␻
eG
.
10
y
10
R = −
͚
8
y
The relevant transition dipoles, not shown explicitly in the
2
2
equations, are assumed to be constants obeying ␮21=2␮10.
This harmonic approximation is consistent with the nearly
equal relative intensities of the two peaks in the pump-probe
2
10
spectra for all the modes. The value of ␮ was taken as
unity in the expressions for the responses. The R3 and R7
diagrams contain an additional e−␣t/T term to account for the
1
faster relaxation of the ␯=2 compared with the ␯=1 level.
For a harmonic oscillator ␣ would be unity. The factors
P͑␯,0͉␯ ,b͒ represent the conditional probabilities that
Ј
given a molecule is in the population state with quantum
number ␯ of the mode at time 0 it will be in a state ͉␯ ͘ at
Ј
time b. These factors may be calculated from the kinetics of
the multiple level system. Each R describes the contribution
to the signal from the evolution occurring in a particular
subensemble of the molecules. The probability P͑0,0͉0,T͒
is taken to be unity. Vibrational energy relaxation T1 is taken
FIG. 2. The measured ͑circle͒ and simulated ͑solid line͒ peak shifts of the
spectrally resolved echo of the N–H stretch vibration as a function of wait-
ing time T. The inset shows the vibrational frequency correlation function.
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064508-4
Ha, Kim, and Hochstrasser
J. Chem. Phys. 124, 064508 ͑2006͒
into account through the multiplication factors, exp͑−͑␶
+t͒/2T1͒. In the vibrational relaxation process other internal
and intermolecular modes may become excited. While the
primary modes in the relaxation pathway would be those that
are strongly anharmonically coupled to the mode being con-
sidered, some of these accepting modes may pass on their
energy to more weakly coupled modes during the experimen-
tal time scale. For the case of N–H only we assume, as in
previous work,10 that levels ͉y͘ are excited by relaxation of
the ␯=1 state. The ͉y͘ states do not have any N–H excitation
but they may be excited to levels ͉y+1͘ by infrared pulses in
the range of 3 ␮m corresponding to the addition of one
quantum of N–H excitation. Possibly there are many states
involved in the relaxation, but they are modeled by two re-
sponse functions R4 and R8. As before10 the state͑s͒ ͉y͘ are
FIG. 3. The power spectrum of the frequency correlation function of the
sum of the two underdamped oscillatory components of N–H.
¯
assumed to have similar anharmonicity ⌬, and therefore
¯
i⌬t
P͑10͉yT͒ei⌬ t = ͑1 − e−T/T ͒e
.
͑13͒
y
1
͚
The sum of oscillatory terms ͑two damped cosine functions͒
is incorporated because the peak shift profile shows oscilla-
tions around T=160 fs, which means that the correlation
function of the N–H frequency must contain an under-
damped oscillation. With these correlation functions and
knowing that the overall rotational motion makes a negli-
gible contribution,10 the linear spectrum was simulated by
Fourier transformation of the linear response function:
y
In the responses the fluctuations in the mode frequency are
labeled as x͑t͒ while the y-state fluctuations are labeled as y.
The real part of the line-shape function g͑t͒ is then defined as
t
␶
1
g ͑t͒ = d␶
xy
d␶ ͗x͑␶ ͒y͑0͒͘,
͑14͒
͵ ͵
1
2
2
0
0
t−t/2T
Rlinear = e−g͑t͒−i␻
.
͑17͒
10
1
which is symmetric in x and y. The fluctuations in ␻ and
10
␻
are assumed to be strictly correlated so the same param-
21
As shown in Fig. 1, the simulation agrees well with experi-
ments except for the small deviation at lower frequency. The
eters were used in gxx for both the ␯=0→␯=1 and the ␯
=1→␯=2 transitions. The parameters in gyy, which describe
the N–H mode dynamics when it is associated with a mode
͉y͘, are the same as those in gxx. It was shown previously10
that frequency fluctuations of the N–H and the ͉y͘ modes are
correlated with a correlation coefficient of 0.51. The third-
order real three-time electric field signal was simulated by
convolution of the response functions Rj with the electric
fields of the laser pulses:
˜
absolute magnitude quantity ͉SR͑␶,T,␻ ͉͒ was obtained from
t
the Fourier transform of the third-order real three-time
electric-field signal and used to simulate and fit the peak
shifts for a set of T values. The best-fit set of 14 parameters
in the correlation function which satisfy both the linear spec-
trum and peak shift profile were found to be a1=10 ps−1,
a2=6.1 ps−1, a3=9.9 ps−1, a4=4.9 ps−1, a5=4.9 ps−1, t1
=0.15 ps, t2=0.86 ps, t3=11 ps, ␶1=0.23 ps, ␶2=0.25 ps,
␻1=80 cm−1, ␻2=100 cm−1, ␾1=1.0, and ␾2=0.85. This
correlation function, shown as an inset in Fig. 2, exhibits a
peak at 240 fs and requires a phase shift. Among the three
exponential components, the time constant of the fastest part
is much faster than the estimates based on the many fewer
data points of the previous experiment. The oscillation
around T=240 fs was not definitively captured in the earlier
report although it was the hint of these oscillations from a
few data points that led to the present study. The oscillatory
feature is well represented by the sum of the two damped
cosine functions having similar amplitudes and damping
time constants.
The power spectrum of the Fourier transform of the sum
of the oscillatory parts is shown in Fig. 3. A power spectrum
peaking at ϳ95 cm−1 is consistent with the results from far-
infrared absorption,34 optical-heterodyne-detected Raman-
induced Kerr effect spectroscopy ͑OHD-RIKES͒,35–37 and
depolarized Rayleigh/Raman spectroscopy.38,39 In these ex-
periments peaks at ϳ100 and ϳ190 cm−1 were observed and
assigned as hydrogen-bond librational motional modes. The
low-frequency vibrational mode near ϳ100 cm−1 appears to
ϱ
ϱ
ϱ
S͑␶,T,t͒ = Re
dt
dt
dt
R ͑t ,t ,t ͒
j 1 2 3
͵ ͵ ͵
͚
1
2
3
ͭ
−ϱ
−ϱ
−ϱ
j
ϫ E3͑t − t3͒e−i␻͑t−t ͒
3
ϫ E2͑t + T − t3 − t2͒e−i␻͑t+T−t −t ͒
3
2
ϫ E1͑t + T + ␶ − t3 − t2 − t1͒e−i␻͑t+T+␶−t −t −t ͒
.
3
2
1
ͮ
͑15͒
The response functions Rj’s in the simulation incorporated
all time orderings that could arise from pulses of duration of
75 fs.
The vibrational frequency correlation function of the
N–H mode is described by the following fitting function with
14 parameters:
͗x͑t͒x͑0͒͘ = a21 exp͑− t/t1͒ + a22 exp͑− t/t2͒
+ a2 exp͑− t/t3͒ + a2 exp͑− t/␶ ͒cos͑␻1t + ␾ ͒
1
1
3
4
+ a2 exp͑− t/␶ ͒cos͑␻2t + ␾ ͒.
͑16͒
2
2
5
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064508-5
Vibrational dynamics in formamide
J. Chem. Phys. 124, 064508 ͑2006͒
FIG. 4. ͑Color͒ The measured ͑left panel͒ and simulated ͑right panel͒ absorptive 2D IR spectra of the N–H stretching mode of the formamide N–H bond
vibration in formamide-D3 as a function of time T. ͓͑a͒ and ͑b͔͒ T=210 fs and ͓͑c͒ and ͑d͔͒ T=600 fs.
correspond to the ϳ95 cm−1 mode observed in the Fourier-
transformed power spectrum of our experiment. The oscilla-
tion in the peak shift demonstrates that the N–H stretch mode
oscillation is modulated by the underdamped intermolecular
hydrogen-bond motion in the liquid.
This effect is the spectral manifestation of the spectral diffu-
sion implied by the correlation function and Eq. ͑14͒.
B. DCONH2 in formamide „HCONH2…
The 2D IR spectra are shown in Figs. 4͑a͒ and 4͑c͒.
Figure 4͑a͒ corresponds to the 2D spectrum measured at T
=0.21 ps and Fig. 4͑c͒ is that at T=0.6 ps. The simulations
of these spectra corresponding to the fitting function given
above are shown in Figs. 4͑b͒ and 4͑d͒, respectively. The
simulated 2D IR spectral shapes show very good agreement
with experiment. Two main peaks which correspond to the
0→1 and 1→2 transitions of the N–H stretching mode are
The linear IR-absorption spectrum ͑solid line͒ of the
C–D stretch vibration near 5 ␮m of DCONH2 in formamide
is shown in Fig. 5 with the background absorption from for-
mamide removed. The C–D absorption band ͑ϳ39 cm−1͒ is
much narrower than that of N–H ͑ϳ180 cm−1͒. The peak
position is shifted to higher frequency compared with forma-
mide in nonhydroxylic solvents40 as a result of hydrogen
bonding. The IR pulses were centered at the C–D peak po-
sition and had sufficient bandwidth to span the fundamental
0→1 and the 1→2 transitions.
observed along the ␻ axis. The diagonal positive peak is the
t
0→1 transition and the negative peak that is shifted to lower
frequency along the ␻t axis by the diagonal anharmonicity is
the 1→2 transition. The elongated diagonal peak at T
=0.6 ps is elliptically shaped, which implies that the inho-
mogeneous broadening of the 0→1 transition is persistent
until T=0.6 ps. However, as the T values get larger from
0.21 to 0.6 ps, the diagonal peaks become more circular.
˜
The profile of the quantity ͉S͑␶,T,␻ ͉͒ vs ␶ at different
t
˜
values of T were obtained by adding the signals ͉͑S͑␶,T,␻ ͉͒͒
t
over the range of ␻ from 2158 to 2178 cm−1. This proce-
t
dure samples only the fundamental 0→1 transition of the
C–D stretch. As shown in Fig. 6, the peak shift obtained by
this procedure decays but shows no oscillation in contrast to
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064508-6
Ha, Kim, and Hochstrasser
J. Chem. Phys. 124, 064508 ͑2006͒
FIG. 6. The measured peak shifts ͑circle͒ of the spectrally resolved echo of
the C–D stretch vibration as a function of waiting time T. The fitted solid
line is the peak shift simulated with the parameters in Table I. The vibra-
tional frequency correlation function is shown in the inset.
FIG. 5. The linear IR-absorption spectrum ͑solid line͒ of the C–D stretch
vibration of DCONH2 in formamide-H3. The dotted line is the spectrum of
the linear-response function ͓Eq. ͑17͔͒.
parameters that best fitted the linear spectrum and the peak
shift were found to be a1=1.2 ps−1, a2=2.3 ps−1, a3
=2.4 ps−1, t1=0.06 ps, t2=0.33 ps, and t3=4.3 ps. The time
constants for the three decay components of the correlation
function are somewhat shorter than those of the N–H mode
in formamide when it is fitted to three exponentials. The
amplitudes representing the distribution of frequencies are
also considerably smaller for C–D than N–H ͑see Table I͒.
The 2D IR spectra measured at T=300 fs and T=2 ps
are shown in Figs. 7͑a͒ and 7͑c͒, and the simulated 2D IR
spectra in Figs. 7͑b͒ and 7͑d͒. The simulated spectral shapes
agree well with the experiments. There are two peaks along
the case of the N–H stretch mode. The peak shift of C–D
does not show any increase in the picosecond region compa-
rable with that reported for N–H, so the cross correlation
pathways involving the ͉y͘ states were not included in the
analysis. In these simulations, the C–D frequency fluctua-
tions are completely characterized by ͗x͑t͒x͑0͒͘ which is
symmetric in x. We assumed that ␻ and ␻ have strictly
correlated fluctuations and the same parameters were used in
gxx for both the ␯=0→␯=1 and the ␯=1→␯=2 transitions.
The third-order real three-time electric-field signal was again
simulated by convolution of the response functions with the
electric fields of the laser pulses ͓Eq. ͑15͔͒. The pulse dura-
tion of 80 fs was used in this numerical convolution. The T1
relaxation time of the C–D vibration was measured to be
10
21
the ␻ axis corresponding to the 0→1 ͑the diagonal positive
t
peak͒ and 1→2 transitions ͑the negative peak͒ of the C–D
stretching mode. The diagonal anharmonicity is estimated to
be 57 cm−1 from the fitting. The main peaks in the 2D IR
spectrum are somewhat elongated along the diagonal at T
=0.3 ps, which implies residual inhomogeneous broadening
of the 0→1 transition. However, as the T value increases
from 0.3 to 2 ps, the diagonal main peaks become nearly
circular as a result of the spectral diffusion implied by the
parameters of Eq. ͑18͒.
˜
1.5 ps from the transient grating experiment, ͉S͑␶=0,␻t;T͉͒
vs T.
The data were fitted by using a frequency correlation
function for the C–D mode consisting of three Kubo
functions:
͗x͑t͒x͑0͒͘ = a21 exp͑− t/t1͒ + a22 exp͑− t/t2͒
+ a23 exp͑− t/t3͒.
͑18͒
The linear spectrum, simulated by Fourier transformation of
the linear-response function using Eqs. ͑14͒ and ͑17͒, is
shown in Fig. 5. It agrees with experiment near the center
frequency of the vibration but there is the slight disagree-
ment at the high- and low-frequency wings. The set of six
C. HCONH2 in bulk formamide
The amide-I IR absorption at 1685 cm−1 in liquid forma-
mide has a bandwidth of 45 cm−1, which is narrower than the
N–H and wider than the C–D transitions. The peak position
TABLE I. The amplitudes and correlation times of each component in the fitted three exponential data for three
IR active modes.
␶
␶
␶
3
1
2
a1
͑ps͒
a2
͑ps͒
a3
͑ps͒
N–H
C–D
10 ps−1
1.2 ps−1
0.004 ps
0.15
0.06
0.02
6.1 ps−1
2.3 ps−1
0.051 ps
0.86
0.33
0.75
9.9 ps−1
2.4 ps−1
0.017 ps
11
4.3
11
CvOa
aThe amplitudes in this row correspond to peak shifts obtained by fitting the data in Fig. 8 to three exponentials
as discussed in the text.
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064508-7
Vibrational dynamics in formamide
J. Chem. Phys. 124, 064508 ͑2006͒
FIG. 7. ͑Color͒ The measured absorptive 2D IR spectra ͑left panel͒ of the C–D stretching mode of the formamide C–D vibration in formamide-H3 as a
function of time T. The simulated spectra are shown on the right panels. ͓͑a͒ and ͑b͔͒ T=300 fs and ͓͑c͒ and ͑d͔͒ T=2 ps.
is shifted to lower frequency by 25 cm−1 compared with
chloroform ͑1710 cm−1͒.40 Such shifts are typical of H bond-
ing of carbonyls in peptides. The IR pulses used in the non-
linear experiments have sufficient spectral bandwidth to
bracket the complete amide-I transition region.
IV. DISCUSSION
A. Fast dynamics of N–H motion
The correlation functions are described as sums of expo-
nentials, each of which represents a Kubo function. These
The two-dimensional data were processed to obtain a
function s͑␶,T͒ that is the integral over all t of the absolute
magnitude of the complete three-time data set S͑␶,T,t͒. The
function s͑␶,T͒ resembles the conventional41 peak shift sig-
nal that is the integral over t of the absolute magnitude
squared of S͑␶,T,t͒. The profile of s͑␶,T͒ is peaked at dif-
ferent ␶ times depending on the choice of waiting time T.
The T dependence of the shift of this peak is plotted in Fig.
8 and reasonably well fitted by three exponentials. The re-
sults are summarized in Table I. Note that for CvO the
correlation function is not determined by a simulation based
on the response functions as was the case for N–H and C–D.
Rather, a representation of the frequency correlation function
was obtained by direct fitting of the peak shift to a sum of
exponentials in the expectation that the parameters obtained
would be associated with different physical processes in the
liquid. The T1 relaxation was also measured by processing
the same data set to yield T1=0.62 ps.
FIG. 8. The measured peak shifts ͑circle͒ of the integrated echo and the fit
͑solid͒ to three exponential functions. The gray line is the residual of the fit.
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064508-8
Ha, Kim, and Hochstrasser
J. Chem. Phys. 124, 064508 ͑2006͒
functions are not proven to be the actual forms of the corre-
lation decay but represent a convenient way to fit the data to
a set of parameters besides suggesting a simple physical in-
terpretation. The reported parameters are a valid representa-
tion of the correlation function if the vibrational responses,
Rj of Eq. ͑15͒, are appropriately defined. Recent works by
Skinner and co-workers have drawn attention to the fact that
this approach is inappropriate for liquid water.42,43 We expect
that the separated time scales may be associated with differ-
ent physical processes. The fastest correlation time in the
frequency correlation function was measured to be 0.15 ps
and its preexponential factor is 10 ps−1, corresponding to a
Gaussian width of 125 cm−1, which is slightly smaller than
the frequency shift of the N–H stretching mode on forming
H bonds to CvO groups. The fastest correlation time cor-
responds to the modulation of the hydrogen-bond distance
and is measured to be close to the fast modulation limit. In
the intermediate region below ϳ1 ps, there are three terms.
One is the exponential decay function having the time con-
stant of 860 fs. The other two terms are the sum of the un-
derdamped oscillation terms having the time constants of
ϳ240 fs. In the previous work, the 860 fs correlation time
was shown to incorporate only a slight angular motion within
a cone angle of 19°. This motion was considered to be re-
lated to the overdamped librational motion or hindered rota-
tion around the intermolecular hydrogen bond. This is an
angular motion over a space that is much smaller than
needed for overall rotation so it is connected with fast angu-
lar fluctuations of hydrogen bond. On the other hand, as
shown in Fig. 3, the power spectrum of the oscillatory parts
͑the sum of two underdamped oscillation terms͒ clearly
shows a peak at ϳ95 cm−1. Other optical spectroscopies
such as far-infrared absorption, OHD-RIKES, and depolar-
ized Rayleigh/Raman spectroscopy have indicated that for-
mamide has low-frequency vibrational modes at ϳ100 and
ϳ190 cm−1.34–39 The underdamped motion in our experi-
ment most likely corresponds to the ϳ100 cm−1 mode,
which was previously assigned as an intermolecular libration
of H-bonded formamide molecules. Ab initio calculations of
formamide clusters assign this frequency to the out-of-plane
rotational motion around an axis through the trans N–H
bond.44,45 In the echo experiment, the correlation function
does not clearly show an ϳ190 cm−1 oscillation. The ab-
sence of this mode may be caused by the limited laser-pulse
bandwidth ͑ϳ190 cm−1͒, but it is possible that the
ϳ190 cm−1 librational intermolecular motions do not signifi-
cantly modulate the N–H stretch coordinate or that the asso-
ciated form responsible for this mode37 has a too low con-
centration in the liquid.
spectral diffusion is considered to be the movement of the
hydrogen-bonded network, this faster correlation decay im-
plies that the environment probed by the C–D mode is con-
siderably more flexible and less tightly coupled than that
sensed by the N–H mode.
The frequency correlation functions of the C–D mode do
not show underdamped oscillatory components. For the un-
derdamped C–D¯O motion to be incorporated effectively
into the C–D vibrational transition frequency, there must be a
strong correlation between the C–D stretching vibration and
the C¯O distance ͑R͒.46–49 The ab initio molecular-
dynamics ͑MD͒ simulations predict a broad distribution of
C–D¯O angles, which is consistent with our
measurement.18 Another reason why oscillatory components
are not seen in the frequency correlation function of the C–D
vibration could be related to the average angle between the
C–D and N–H bonds. The 100 cm−1 librational motion origi-
nates from the rotational motion in the out-of-plane direction
around an axis that is close to the trans N–H bond:44,45 this
axis is almost perpendicular to the C–D bond. Therefore, this
librational mode may be only weakly coupled to the C–D
stretching mode.
The C–D and N–H stretching vibration modes are, in
principle, coupled through the intramolecular potential.
Therefore, if the N–H bond has low-frequency components
due to its coupling to the environment these motions might
be expected to show up in other formamide modes that are
anharmonically coupled to N–H. We employed density-
functional theory ͑DFT͒ at the B3LYP/6-31+G** level to
examine the modes and mode couplings in formamide using
the previously described approach.50 The potential-energy
distribution in the harmonic force field shows that 99% of
the C–D stretch normal mode is localized on the C and D
atoms. Furthermore, a computation of the diagonal anharmo-
nicity of the C–D mode shows that 69.1 cm−1 out of the total
75 cm−1 is the intrinsic anharmonicity of the C–D motion.
Coupling to the CvO mode contributes 3 cm−1, while cou-
pling to the N–H contributes only 2ϫ10−4 cm−1 to the C–D
anharmonicity. These results strongly suggest that perturba-
tions on the N–H motion by the environment are not likely to
be sensed by the C–D bond vibration, which is very strongly
isolated from N–H in these computations. Our experiments
are consistent with this prediction.
C. Relation of the C–D frequency correlation function
to liquid structure and dynamics
The C–D stretch frequency is shifted to higher frequency
on hydrogen bonding.51 This behavior is opposite to that
found for N–H and O–H but similar to that found for some
ions. A possible explanation52–54 of such behavior is that the
electric field created by the acceptor atom in a hydrogen
bond gives rise to a force that tends to lengthen or shorten
the X–H bond depending on whether the dipole derivative of
the bond is parallel or antiparallel to the field.
The preexponential factor ͑a1͒ of the fastest, motionally
narrowed, part of frequency correlation function of the C–D
stretch mode is 1.2 ps−1. From this preexponential factor, we
can estimate the corresponding inhomogeneous width to be
B. Comparison of the frequency correlation functions
of the N–H and C–D modes
The frequency correlation functions of the N–H and
C–D vibrators yield quite different parameters when fitted to
three exponentials ͑see Table I͒. In each time regime identi-
fied by the fitting, the C–D stretching mode correlation times
are faster than those of N–H, indicating the faster spectral
diffusion for the C–D stretch band. Since the main source of
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064508-9
Vibrational dynamics in formamide
J. Chem. Phys. 124, 064508 ͑2006͒
−1
ͱln 4/␲c=15 cm . As in the case of the N–H oscillator,
that the transition dipole for C–D is the same within 10% for
formamide not H bonded ͑in CHCl3͒ and H bonded in for-
mamide liquid.
a1
this value is not much smaller than the frequency shift of the
C–D stretching mode on forming a hydrogen bond to the
CvO group. The C–D frequency for formamide shifts by
21 cm−1 between CHCl3 and the neat liquid. With this inter-
pretation of the amplitude, the fast correlation time corre-
sponds to the modulation of H-bond distance. The much
smaller frequency shift of the C–D compared with the N–H
stretch mode on forming a hydrogen bonding is consistent
with the hydrogen-bonding strength of the C–D to CvO
being much weaker. On the other hand, the correlation time
of the fastest part of the C–D stretch frequency correlation
function is faster than that found for N–H and it is in the fast
modulation limit.
The rotational relaxation of the N–H bond in liquid for-
mamide is dominated by an 11 ps component.10 The contri-
butions to the overall dephasing on the ca. 1 ps time scale
when angular motions are restricted can be assessed by their
relative amplitudes in the frequency correlation function. The
intermediate time scale contributes 39% to the t=0 ampli-
tude of the C–D frequency correlation function but only 23%
to that of the N–H stretch mode. The spectral diffusion is
faster and more extensive for C–D than for N–H during this
intermediate time scale when the overall motions are
strongly restricted. We conclude that the changes in liquid
state structure in the immediate vicinity of the C–D bond are
more significant than those around N–H, consistent with
there being fewer C–D hydrogen bonds.18 In the case of the
N–H bond, the ca. 1 ps motions were considered to mediate
the vibrational relaxation by fluctuating the force exerted on
the N–H bond.10 However, for the C–D bond, the T1 relax-
ation time is 1.5 ps, which is much slower than the correla-
tion time of the intermediate time scale component. It is
suggested that the weak hydrogen bonding of C–D generates
significantly smaller forces than exerted on N–H by carbonyl
groups.
The slowest frequency correlation component of the
N–H bond was assigned as the breaking and making of the
hydrogen-bonded structure. For the C–D stretching vibration
motion, the slowest frequency correlation time is 4.3 ps,
which is one-third of the time found for the N–H bond.
Clearly the making and breaking dynamics of the C–D hy-
drogen bond is much more rapid than that of N–H.
The positive diagonal signals ͑the fundamental 0→1
transitions͒ of the experimental 2D IR spectra of C–D agree
well with the simulations at T=300 fs, indicating that the
distribution of frequency fluctuations of C–D is almost
Gaussian during this early period ͑see Fig. 7͒. However, at
T=2 ps the high-frequency part of the 2D IR spectrum has a
narrower bandwidth than seen in the simulated spectrum.
This effect suggests that the relaxation rates are slightly fre-
quency dependent. The high-frequency part of the C–D vi-
brational spectrum corresponds to the more hydrogen-
bonded species, for which the spectral diffusion process may
be slower than that of the more free species that dominate the
low-frequency edge of the band. Another possible reason for
the variations in shape could be the frequency dependence of
the transition dipole moment.42,43 However, we have shown
D. Vibrational dynamics of amide-I mode
in bulk formamide
For the amide-I transition the response functions are ex-
pected to be more complicated than those for a single oscil-
lator, since it is obvious from the 2D IR spectra and linear IR
spectra that there is significant mode coupling between dif-
ferent molecules.55–57 However, the decay of the echo peak
shift is expected to yield a qualitative representation of the
frequency correlation function. We note that these amide-I
experiments were not fitted to simulations using the response
functions as in the N–H and C–D examples. The amide-I
peak shift data were directly fitted to a sum of three expo-
nentials. The most slowly varying part of the CvO peak
shift data corresponds approximately to the same structure
breaking time that was observed for the N–H mode and is
consistent with the orientational dynamics of the N–H bond
͑see Table I͒. Furthermore, an intermediate time scale of ca.
1 ps is observed for both N–H and CvO. During this 1 ps
period the orientational order is largely maintained and so
hydrogen-bond fluctuations within a given set of structures
are the most obvious interpretation.
The fastest part of amide-I frequency correlation func-
tion makes only a small contribution to the peak shift profile
suggesting that the relevant distribution is motionally nar-
rowed. We know from fitting to the response functions that
motionally narrowed contributions have a small influence on
the correlation function. The intermediate time scale process
has a peak shift of ϳ10 fs for N–H. From the fact that the
peak shift is inversely related to amplitude factor in the cor-
relation function,58 it can be concluded that the distribution
of N–H frequency fluctuations corresponding to the interme-
diate time scale is about five times wider than for CvO.
These relative magnitudes track the spectral shifts on hydro-
gen bonding of CvO ͑30 cm−1͒ and N–H ͑150 cm−1͒. The
correlation times of the intermediate process are very similar
for CvO ͑0.75 ps͒ and N–H ͑0.86 ps͒, suggesting that the
same H-bond dynamics are involved in the spectral diffusion
of both modes. The C–D bond peak shift does not exhibit
this 0.8 ps process, showing clearly that it is located in an
environment that dephases the mode more rapidly ͑0.3 ps͒.
However, the intermediate time scale ͑0.3 ps͒ is still domi-
nant in the peak shift decay of C–D suggesting that C–D is
also involved in hydrogen-bonding interactions.
V. CONCLUSION
The vibrational frequency correlation function of the
C–D stretch mode decays in three time regimes ascribed to
the modulation of the H-bond distance, the overdamped li-
bration of the intermolecular hydrogen bond, and the overall
structure breaking and making. The correlation times for the
C–D stretch mode obtained by fitting to three exponentials
are faster than those of the N–H mode, implying that the
liquid structure around the C–D bond is less rigid and the
related motions of the C–D bond are relatively faster than
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064508-10
Ha, Kim, and Hochstrasser
J. Chem. Phys. 124, 064508 ͑2006͒
19 G. A. Jeffrey, An Introduction to Hydrogen Bonding ͑Oxford University
Press, New York, 1997͒.
that of N–H. Underdamped oscillatory behavior having the
character of the low-frequency hydrogen-bond intermolecu-
lar librational motion was apparent only for the N–H stretch
mode. In the case of the amide-I ͑carbonyl͒ mode, the fre-
quency distribution of the fastest component is motionally
narrowed and the intermediate and slow components have
similar characteristic time constants to those of N–H, indi-
cating that they are both caused by the strong intermolecular
hydrogen-bonding interaction. Liquid formamide is envis-
aged as a N–H¯OvC hydrogen-bond network, reinforced
by the weaker C–H¯OvC hydrogen bonds. The confor-
mational changes around the C–H bond are the most facile.
These structural conclusions are consistent with recent ab
initio dynamics calculation.18
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