Tuning Molecular Vibrational Energy Flow within an Aromatic Scaffold via Anharmonic Coupling

Article abstract of DOI:10.1021/acs.jpca.9b08010

From guiding chemical reactivity in synthesis or protein folding to the design of energy diodes, intramolecular vibrational energy redistribution harnesses the power to influence the underlying fundamental principles of chemistry. To evaluate the ability

Full text of DOI:10.1021/acs.jpca.9b08010

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A: Spectroscopy, Molecular Structure, and Quantum Chemistry
Tuning Molecular Vibrational Energy Flow within
an Aromatic Scaffold via Anharmonic Coupling
Andrew J. Schmitz, Hari Datt Pandey, Farzaneh Chalyavi, Tianjiao Shi, Edward
Emmett Fenlon, Scott H Brewer, David M. Leitner, and Matthew J. Tucker
J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.9b08010 • Publication Date (Web): 17 Nov 2019
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The Journal of Physical Chemistry
Tuning Molecular Vibrational Energy Flow within an Aromatic Scaffold via Anharmonic
Coupling
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Andrew J. Schmitz, a Hari Datt Pandey,c Farzaneh Chalyavi,a Tianjiao Shi,b Edward E. Fenlon,b Scott
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H. Brewer,b David M. Leitnera and Matthew J. Tucker*,a
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aDepartment of Chemistry, University of Nevada, Reno, Nevada 89557, United States
bDepartment of Chemistry, Franklin & Marshall College, Lancaster, Pennsylvania 17604-3003, United States
cDepartment of Chemistry, University of California Riverside, California, 92521, United States
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KEYWORDS. Vibrational Energy Flow, 2D IR, Anharmonic Coupling, Probe Pairs, Fermi Resonance
ABSTRACT: From guiding chemical reactivity in synthesis or protein folding to the design of energy diodes, intramolecular
vibrational energy redistribution harnesses the power to influence the underlying fundamental principles of chemistry. To
evaluate the ability to steer these processes, the mechanism and timescales of intramolecular vibrational energy
redistribution through aromatic molecular scaffolds have been assessed by utilizing two-dimensional infrared (2D IR)
spectroscopy. 2D IR cross peaks reveal energy relaxation through an aromatic scaffold from the azido- to the cyano-
vibrational reporters in para-azidobenzonitrile (PAB) and para-(azidomethyl)benzonitrile (PAMB) prior to energy relaxation
into the solvent. The rates of energy transfer are modulated by Fermi resonances, which are apparent by the coupling cross
peaks identified within the 2D IR spectrum. Theoretical vibrational mode analysis allowed determination of the origins of the
energy flow, the transfer pathway, and a direct comparison of the associated transfer rates, which were in good agreement
with the experimental results. Large variations in energy transfer rates, approximately 1.9 ps for PAB and 23 ps for PAMB,
illustrate the importance of strong anharmonic coupling, i.e. Fermi resonance, on the transfer pathways. In particular,
vibrational energy rectification is altered by Fermi resonances of the cyano- and azido- modes allowing control of the
propensity for energy flow.
ambiguity in VER, in particular IVR, arises because the
energy transfer occurs from one bright mode to another
mode via multiple dark states. This process creates
difficulties in determining the rate and exact pathway of
energy transfer within the molecular system.22 Since
intramolecular pathways are often more favorable, there
are limitations in how much energy will propagate through
the solvent. This complicates intermolecular energy
transfer studies. However, Zheng and coworkers have
developed methods to design molecules that decouple the
modes making intermolecular vibrational energy transfer
the preferred route.22, 43
Introduction
For the past two decades two-dimensional infrared (2D
IR) spectroscopy has become the premiere ultrafast
technique to characterize vibrational dynamics. It has been
used to study a wide array of systems, such as structural
dynamics in biomolecules, energy transfer in materials,
population exchange, and properties of ion channels in cell
membranes. 1-7 Additionally, the localized solvent dynamics
in heterogeneous solutions, which play an important role in
reactivity, are also determined through spectral diffusion
observed using 2D IR spectroscopy.
8-12
Another valuable
characteristic of 2D IR is the ability to detect interactions
between multiple vibrational transitions within a molecular
system. For example, the presence of vibrational coupling,
i.e. mechanical and/or dipole-dipole coupling5, 13-14, is
oftentimes monitored through the presence of cross peaks
in the 2D IR spectrum at the earliest evolution or waiting
times.15 1, 5, 16-18 Cross peaks also can appear as a function of
later waiting times. These spectral features measure the
dynamics resulting from vibrational energy relaxation
(VER). In this case, the two modes are delocalized within a
molecular system causing thermal energy flow via VER
within the molecule as intramolecular vibrational energy
redistribution (IVR) or through the solvent as
intermolecular vibrational energy transfer. 19-23The overall
process is delayed since the energy absorbed from one
detected mode must propagate within the molecule until
VER plays a vital role in many physical and chemical
processes, such as chemical reactions where selected
vibrational energy flow influences the reaction coordinate
aiding in synthetic chemistry,20,
molecular
22, 40, 43-46
electronics and electrochemistry,38-39, 47-49 measuring three
dimensional structures of molecular systems,24,
and
50-53
energetic properties of transition metals.20-21, 41, 48
VER
also influences protein functions through hydrogen
bonding along protein backbones,32,
formation of
54
secondary and tertiary structures,34 and allosteric
communication.55-56 By learning ways to control VER, the
ability to influence these processes in a desirable way
becomes possible.
Several recent 2D IR studies have expanded our
understanding of VER and some of the factors guiding its
pathway. The strong carbonyl mode in transition metal
complexes was used by Kubarych and co-workers to
investigate the effects of water on the relaxation of the
carbonyl.57 The IVR rate of ruthenium versus iron
reaching the other detected mode, reflected by the intensity
growth timescale.19,
VER have been studied both
21, 24
theoretically25-34 and experimentally, where nonlinear
techniques, such as IR pump Raman probe and 2D IR, have
provided great insights into the rates of VER, direction of
heat flow, and diffusive/ballistic transport.19, 21, 35-42 Despite
the vast knowledge acquired from such studies, the overall
process of VER still remains challenging to understand. The
complexes were compared,21
and the influence of
hydrogen bonding on IVR rates was determined.58 Other
studies have captured the energy transfer through
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Figure 1. FT IR spectra of A) PAMB and B) PAB in THF and absorptive 2D IR spectra of C) PAMB and D) PAB in in THF at T = 3
ps.
transition metals suggesting a heavy atom can influence
efficiency, direction of IVR, and the contribution of solvent
on IVR.41, 59-61 Zheng and co-workers have contributed an
abundance of work on intermolecular energy transfer via
2D IR, including the determination of energy transfer rates,
manifolds, our 2D IR measurements detected the presence
of some dipole-dipole coupling and extensive IVR. Herein,
the details of the coupling processes for these systems are
determined through a combination of experimental and
theoretical methods showing that the molecular energy
flow is tuneable within this aromatic scaffold via alteration
of the anharmonic mode coupling. Our approach uses a
model system to uncover the coupling manifold between
the azido- transition, several combination bands, and the
cyano- transition to illustrate how the transfer pathway can
be influenced to guide the energy flow. This information is
necessary for the design of molecular electronics.
Furthermore, by limiting the IVR pathway to one direction,
molecular selectivity in synthesis and biological function
may become possible.
distances, and orientations between two modes on different
40, 43, 47, 49-51
molecules.22,
Rubtsov and co-workers have
established methods of coupling involving IVR coined
relaxation-assisted two-dimensional infrared spectroscopy
(RA 2DIR).19 Rubtsov developed his methodology based on
observations of IVR between a variety of vibrational mode
62-64 65
pairs,24,
including CN/CO, CN/amide I modes, and
N3/CO, in several molecular scaffolds with varying
configurations and bond distances between the transitions.
His work has uncovered a correlation between energy
53, 63
transfer rate and bond length,49,
temperature
dependence of energy transfer rates,65 and cross peak
enhancement.48, 62-63
Experimental
Synthesis of PAB and PAMB
Our group, as shown in previous work, has an interest in
studying the vibrational dynamics of the azido- (N3) and
4-azidobenzonitrile was prepared by a literature method.69-
66
cyano- (CN) transitions within biological systems.4-5,
70
Complete synthetic details are given in the Supporting
These modes have been established as valuable infrared
probes due to their sensitivity to their local environment,
small size to limit perturbation to the molecular system, and
the large transition dipole strength of azides, which
increases the signal-to-noise ratio. Lastly, in biological
Information. In short, 4-aminobenzonitrile was diazotized
with nitrous acid and then treated with sodium azide to give
PAB in 85% yield. 4-(azidomethyl)benzonitrile was
prepared by the SN2 reaction of sodium azide with 4-
systems, these transitions are spectrally isolated from other
bromomethyl
benzonitrile
to
provide
4-
67-68
congested regions of the IR spectrum.4-5,
Recently,
(azidomethyl)benzonitrile in 83% yield. This is essentially
the method of Chakraborty et al.,71 except that aqueous
methanol was used as the solvent instead of DMSO.
these transitions have been placed in tandem on molecular
systems revealing dipolar coupling for structural studies
within different biological scaffolds.4-5 To further examine
the coupling mechanisms of these probe pairs, several
molecular models were designed and synthesized,
including para-azidobenzonitrile (PAB) and para-
(azidomethyl)benzonitrile (PAMB). Within these testing
Linear and 2D IR Measurements
FTIR and 2D IR measurements were performed in a
solution of tetrahydrofuran (THF) with a concentration of
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approximately 30 mM using CaF2 windows in a Harrick DFT/B3LYP/6-31G-level
calculation,
followed
by
1
2
3
4
sample cell with a path length of 50 μm. Linear IR
measurements were performed using a Nicolet 6700 FTIR
spectrometer.
DFT/B3LYP/6-31+G* using the same solvent model for the
SCRF. Finally, the geometry, Hessian, normal modes,
frequencies, and anharmonic constants were calculated
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Table 1. Theoretically determined most probable
combination bands giving rise to the Fermi resonance
and contributing to the energy transfer pathway from the
azido- to the cyano- mode. The third order coupling
훷
TFR
constant, αβγ, third order resonance parameter,
,
αβγ
훥휔
IVR rate, resonance width,
listed.
, and localized region are
2D IR experiments were performed using Fourier-
transform limited 80 fs pulses at a center wavelength of
4550 nm. Three pulses of wave vectors k1, k2, and k3 with
energy of ~1 μJ were incident on the sample generating an
echo signal in the direction ks=-k1+k2+k3. Pulse order of
123 and 213 produced rephasing and nonrephasing,
respectively. Both rephasing and nonrephasing were
properly phased and combined creating the absorptive
spectrum. Spectral changes over time were observed by
varying the waiting time (TW) between the second and third
pulse from 0 to 12 ps. Appropriate Fourier transforms
along the coherence (τ) and detection (t) axes were
performed and plotted as ωτ vs ωt.5
Figure 2. Vibrational transitions contributing to energy
transfer pathway from the azido- to the cyano- mode (top)
PAB molecule and (middle and bottom) PAMB molecule. The
phases and relative size of each arrow are accurate, but the
absolute sizes of the arrows have been exaggerated to clearly
distinguish the amplitude of the displacements of different
atoms.
using DFT/B3LYP/6-31+G**. An ultrafine integration grid
for the two-electron integral calculations with accuracy
−13
10
and very tight convergence criteria were applied
throughout the electronic structure calculation process. All
the DFT calculations were carried out using the Gaussian-
09 computational package. In order to determine a stable
third order coefficient, it was necessary to use the above-
mentioned calculation protocol. The CPCM model for the
THF solvent was utilized at each level of theory.
The rate of relaxation of excess energy in the azido- and
cyano- modes is mediated by the anharmonic interactions
with other modes of the molecule and interactions with the
solvent. The relaxation rate (W) was estimated in terms of
third order anharmonic interaction with a Fermi’s golden
rule calculation for excess energy in mode α with frequency
ωα, which can be expressed as the sum of quantities “decay”
Computations
The initial geometry of each compound was constructed
using the Avogadro visualization package and optimized
using molecular mechanics (MM) with the General Amber
Force Field (GAFF). The MM optimized geometry was
introduced into
optimization, followed by
a
semi-empirical method (PM6)
Hartree-Fock level (HF)
a
calculation with the 6-31G basis set applying the conducting
polarizable continuum model (CPCM) THF model for the
self-consistent reaction field (SCRF). The HF optimized
geometry was taken as an initial structure for
a
푊 = 푊 + 푊
and “collision” terms,
where25-31, 72-73
푑
푐
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|
|2(
)(
)
휙훼훽훾 1 + 푛훽 + 푛훾 훤훼 + 훤훽 + 훤훾
ℏ
further analysis. For the PAMB compound, the IR spectrum
∑
(
)
푊푑 휔훼
=
=
(1a)
(1b)
1
16휔훼
{
}
훽,훾
1
2
3
4
5
6
7
8
2
2
(
)
(
)
훤훼 + 훤훽 + 훤훾
is quite similar to that of PAB with only a change in the
relative intensity of the spectral features. The azido- and
cyano- transitions are located at 2103 cm-1 and 2230 cm-1,
respectively. In the spectral profile of PAMB, the transition
observed at 2130 cm-1 is likely due to the aforementioned
Fermi resonance. However, the intensity of this vibrational
transition is much weaker when compared to the Fermi
transition of PAB. The single carbon atom separation
between the azido- group and the benzene ring is likely
decoupling the ring modes of benzene and the azido-
transition.78
The experimental relative IR intensity ratio between the
azido- to cyano- stretches was determined to be 3.4:1 and
4.7:1 for PAMB and PAB, respectively. The observed trend
in the intensity ratios is in agreement with with the DFT
calculated trend for PAMB (5.9:1) and PAB (8.8:1).
Discrepancies in the absolute intensity ratios are in part due
to the effect of the intensity mixing caused by the Fermi
resonance coupling within the experiment.
휔훽휔훾 휔훽 + 휔훾 ― 휔훼
+
4
|
|2(
)(
)
휙훼훽훾 푛훽 ― 푛훾 훤훼 + 훤훽 + 훤훾
ℏ
∑
(
)
푊푐 휔훼
1
8휔훼
{
}
훽,훾
2
2
(
)
(
)
훤훼 + 훤훽 + 훤훾
휔훽휔훾 휔훾 ― 휔훽 ― 휔훼
+
4
훤훽
is the damping rate of mode β due to coupling to other
푛훼
modes and to the environment,
is the Boltzmann
9
훼
훷
population of mode , and αβγ, are cubic anharmonic
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constants. Since the vibrational relaxation rate that is
calculated influences the value of , this parameter was
훤
― 푊푑
calculated self-consistently solving the equation
훤훼
― 푊 = 0
using a Newton-Raphson scheme iteratively as
detailed in Ref. 73. Resonant coupling can be characterized
by the relative magnitude of the cubic anharmonic constant
푐
훥휔
to the frequency mismatch of the coupled modes(
+ 휔훽 ― 휔훼 훥휔
is also referred to as the
)
|
|
defined as
.
휔훾
resonance distance. The ratio of the magnitude of the
anharmonic coupling to the resonance distance is the third-
order Fermi resonance parameter for the triple of modes
αβγ, defined by,
Theoretical calculations predict multiple transitions
that possibly gives rise to the Fermi resonance present in
the linear IR spectrum. The most probable ones are listed
in Table 1 and shown in Figure 2 (see SI for more details).
The overall coupling strength increases when the
훷αβγ
=
TFRαβγ
(2)
|
|
훥휔
훷αβγ
magnitude of the third order coupling constant,
,
increases and/or when a decrease in the resonance
distance, Δω, is observed resulting in a larger value of the
third order resonance parameter, TFRαβγ,. Typically, if
Modes are resonant when TFR is of order 1 or larger. The
damping rates that appear in Eq. (1) have contributions
from coupling to solvent. The damping rate for a vibrational
TFRαβγ≥1, αβγ>10 cm-1 and Δω<10 cm-1 then the overall
훷
mode due to solvent coupling is taken to be roughly 0.2 ps-
coupling strength will be significant enough to result in an
accidental resonance between the combination bands and
the fundamental transition, i.e. a Fermi resonance. . Based
on these criteria, it is clear to see from Table 1 that the TFR
1
,
33, 38 and we add to the overall damping rate for the triple
of modes in Eq. (1) the value 0.6 ps-1 to account for solvent
coupling. We note that we found in earlier work on solvated
cyanophenylalanine that the results of Eq. (1) do not change
significantly using a range of values from 0.2 ps-1 to 1.5 ps-1
for the contribution of coupling to the solvent.73
is largest (~1.9) between the azido- mode and
a
combination band suggesting these modes contribute to the
Fermi resonance in PAB and further substantiate the other
strong transitions observed in the IR spectrum due to
anharmonic coupling and intensity mixing. On the other
hand, the TFR for PAMB is smaller (~0.8-0.95) suggesting
the anharmonically coupled modes become somewhat
decoupled due to the separation of the ring and the azido-
group. This lower TFR further explains the observed much
weaker IR intensity of the coupled ring modes within the
PAMB compound.
In particular, the calculations show that the azido-
transition at 2193 cm-1 in PAB couples with a combination
band of two modes localized on the ring with vibrational
frequencies at 1327 cm-1 and 839 cm-1. Even though the
resonance distance is larger than 10 cm-1, the coupling
strength is sufficiently large to support the presence of a
Fermi resonance. As seen in Table S1, the next highest TFR
is <0.3 indicating that no other combination bands
contribute to the energy pathway.
Results and Discussion
Linear IR
FTIR and 2D IR measurements were performed on para-
(azidomethyl)benzonitrile
(PAMB)
and
para-
azidobenzonitrile (PAB) in THF. Four transitions are
observed in the linear IR spectrum for PAMB and PAB as
shown in Figure 1A and 1B, respectively. For the PAB
compound, the asymmetric stretch of the azido- transition
appears at 2106 cm-1 and the cyano- stretch is located at
2228 cm-1. Another transition at 2138 cm-1 arises from a
Fermi resonance that has been reported with the azido-
74-76
reporter in other ring systems in various solvents.67,
The origin of this transition is further supported by
anharmonic frequency computations suggesting only two
transitions within this spectral region. A fourth transition is
observed at 2200 cm-1 resulting from an intrinsically weak
77
solvent band of THF,
and thus it is disregarded from
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with different ring modes of similar frequency than those
1
2
identified.
3
4
5
2D IR measurement
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The 2D IR spectra of PAMB and PAB at waiting time T=3
ps, are shown in Figure 1C and 1D, respectively. Akin to the
휈 = 0→휈 = 1
linear IR spectrum, the azido- and cyano-
transitions appear along the diagonal at ωt=ωτ=2106 cm-1
(peak 1’) and ωt=ωτ=2228 cm-1 (peak 3’) for PAB,
respectively. Similar transitions for the PAMB compound
are observed along the diagonal at ωt=ωτ=2103 cm-1 (peak
1) and ωt=ωτ=2230 cm-1 (peak 3). A third transition is
observed in the 2D IR spectrum of PAB along the diagonal
(peak 9’). This transition is due to a Fermi resonance as
indicated by the strong anharmonic coupling cross peaks
shown by the rectangular boxes. As mentioned above, it has
been shown that the azido- mode strongly couples with a
combination of ring mode transitions resulting in a Fermi
resonance. Although little evidence of Fermi resonance
peaks is present along the diagonal in the 2D IR spectrum of
PAMB, a slight elongation of the positive-going azido-
transition towards higher energy in ωt, hilighted by the
black rectangle in Figure 1C, is observed. These weak cross
peaks suggest some coupling is present between the azido-
mode and a transition not detectable in the 2D IR spectrum.
The absence of a diagonal peak further demonstrates that
the fundamental azido- mode is much less coupled to the
combination of ring modes.
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Figure 3. Population decays of the cyano- and azido-
transitions obtained from 2D IR spectrum for A) PAB and B)
PAMB. The lines represent fits of the exponential decays of
the azido- and cyano-.
The negative contours red shifted along the ωt axis
휈 = 1→휈 = 2
represent
transitions that occur because of
The
the anharmonic nature of the oscillations.
anharmonicities of the cyano- transition are comparable for
both PAMB and PAB at 27±2 cm-1 (peak 4) and 30±3 cm-1
(peak 4’), respectively. The azido- transition for both PAMB
and PAB have anharmonicities of approximately 20 cm-1 as
In the case of the PAMB compound, the computations
determine two possible combination bands that likely
couple with the azido- transition at 2164 cm-1. The first
combination band in resonance with this transition is
comprised of a methylene+ring-+azido (1316 cm-1) mode
and a ring+methylene (835 cm-1) mode. The second
combination band is comprised of a methylene (1217 cm-1)
mode and ring+methylene (956 cm-1) mode. The former has
the highest third-order coupling constant (~14 cm-1), while
the second possibility has the highest TFRαβγ of ~0.94,
suggesting a possible weak Fermi resonance. These findings
are supported by the presence of anharmonic coupling
cross peak found in the 2D IR measurements mentioned
below. Regardless, the lower magnitudes of the TFR for
both cases illustrate that the azido- mode is significantly
decoupled from the ring compared to the PAB compound.
No other combination bands have a TFR >0.3 (as shown in
Table S1) indicating no other major contribution to the
energy pathway. It should be noted that although there is
confidence in the identification of the nature of the coupled
modes, the actual resonances could be due to interactions
shown by peaks 2 and 2’ above.
The waiting time
dependence shows that the peak position of the negative
band blue shifts. At T=0 the negative-going transition in
PAMB is located at ωt=2068 cm-1 and decays almost
completely at T ≈2 ps. At T =200 fs a second negative band
appears at ωt=2084 cm-1 and persists. The azido- transition
for PAB shows similar behavior for the negative transition,
located at ωt=2077 cm-1 during early times. This transition
disappears much faster, revealing a second transition at
ωt=2088 cm-1. The presence of dual peaks in the
휈 = 1→휈 = 2
transition has been reported by Gai and co-
workers and described as a corresponding combination
band (see SI).79 As shown in Figure 1C and 1D, that there is
an elongation with the negative contour of the cyano- peak
compared to the positive band suggesting that the cyano-
mode could be experiencing a similar effect.
In addition to the decaying cross peaks due to the Fermi
resonance (black rectangles), several other cross peaks
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grow in as a function of waiting time. These cross peaks,
.
1
labelled as 5-8 for PAMB and 5’-14’ for PAB, result from
intramolecular vibrational energy redistribution from one
fundamental mode to at least two anharmonically-coupled
combination bands of the ring until reaching the final
fundamental mode. In PAMB, the energy transfer from the
azido- transition to cyano- transition are shown by the cross
peaks 5 and 6. The cross peaks 7 and 8 indicate the
possibility of back transfer from cyano- to azido-. Similarly,
cross peaks 5’-8’ represent the same processes in the PAB
compound. The aharmonicities between peaks 5 and 6 and
peaks 5’ and 6’ is ~12 cm-1. In PAB, cross peaks 11’-14’ also
arise at the intersection of the Fermi and cyano- transitions.
These resonances further indicate the modes involved in
the vibrational energy transfer pathway. The cross peaks
will be analysed and discussed in detail below.
2
3
4
5
6
7
8
9
N3 to CN Cross peaks
Several cross peaks in the 2D IR spectrum play an important
role to expose the energy transfer pathway and the transfer
rate. The cross peaks between azido- mode to the cyano-
mode, observed at {ωt, ωτ}={2230, 2103} cm-1 in PAMB and
{ωt, ωτ}={2228, 2106} cm-1 for PAB, labelled as 5 and 5’ in
Figure 1, respectively, capture the essence of the processes
taking place in the model compounds (Figure 1). At the
earliest waiting times, a cross peak is observable, especially
for PAMB, suggesting dipolar coupling is occurring (See
supporting information). Then, these cross peaks exhibit a
growth and a decay in intensity due to IVR from the azido-
to cyano- transitions. Although somewhat counterintuitive,
the uphill energy flow is possible if the energy separation
between vibrational modes is less than kBT, where kB is the
Boltzmann constant and T is the temperature.58 Assuming
room temperature (~293 K), the value is ~200 cm-1 and the
energy separation between the two oscillators is ~130 cm-
1, suggesting uphill IVR energy flow is allowed. On the other
side of the diagonal, the cross peaks, labelled 7 and 8 and 7’
and 8’, appear at T>0, but exhibit an initial lag phaser
suggesting some IVR from the cyano- to the azido- mode
(Figure 1). The overall decay of these cross peaks
represents a combination of the initial transfer followed by
the thermal cooling of the azido- mode.81 Since the cross
peaks exhibit only an initial lag phase representing the
initial growth before decaying, the vibrational energy flow
within these molecules is somewhat masked for the
backward transfer. Since the cross peak amplitudes are
highly dependent on the characteristics of the two
interacting modes, any growth in the cross peaks is likely
masked by the significant loss of population in the probed
state, the weak dipole moment of the cyano- group, and lack
of strong anharmonic coupling with the ring. For more
details on the dynamics of cross peaks of back transfer, see
the supporting information.
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60
CN and N3 Vibrational Lifetimes
The vibrational lifetimes were determined by
휈 = 0→휈 = 1
measuring the peak intensity of the
transitions as a function of the waiting time T. The
vibrational lifetime of cyano- transition for PAB and PAMB
were fit using a single exponential decay, having a decay of
3.69 ± 0.15 ps (Fig. 3A) and 3.87 ± 0.08 ps (Fig. 3B),
respectively.
A bi-exponential fit was utilized to determine the azido-
transition lifetime decays of T10=0.69 ± 0.02 ps and
T’10=7.04 ± 0.07 ps with amplitudes of A10=0.85 ± 0.01 and
A’10=0.15 ± 0.01 for PAMB (Fig. 3B). PAB (Fig. 3A) has
similar lifetimes of T10=0.83 ± 0.12 ps and T’10=7.33 ± 1.40
ps with amplitudes of A10=0.71±0.06 and A’10=0.29±0.05.
The bi-exponential decay is often detected in transitions
around this spectral region due to vibrational coupling
between the bright mode and spatially close dark modes
within the molecule. The fast component is from an
equilibrium being reached for the coupling of the bright
mode and a dark mode. The longer lifetime is from the
vibrational population decay into the ground state.47, 60, 80
The intensity decay of the Fermi transition along the
diagonal also follows a similar trend to that of the azido-
transition.
Overall, these experimental vibrational lifetimes are in
reasonable agreement with the values calculated with Eq.
(1). For PAB, the theoretical values are 4.46 ps and 0.34 ps
for the cyano- and azido- transitions, respectively. For
PAMB, calculated cyano- transition was determined to be
4.13 ps and the azido- vibrational lifetime was 1.15 ps. It
should be noted that the experimental vibrational lifetime
has two components. However, the fit of the data indicates
that the most dominant contribution is from the short
component (~85% for PAMB and ~70% for PAB). It is the
short component that is captured by Eq. (1), which provides
an estimate to the rate constant for the single-exponential
decay.
At T=0 no positive cross peak is present in the
spectrum of PAB, but instead an elongated negative peak is
observed from ωt≈2190 cm-1 to ωt≈2240 cm-1 as shown in
Figure S2. The positive transition finally appears after T≈1.6
ps. There are several possible reasons for the origin of the
elongated negative-going transition. One possibility is that
the transition originates from fast vibrational energy
transfer from the azido- mode to dark transitions found
within the ring. Oftentimes in the IVR process, a negative
cross peak arises due to dark modes coupling with the
transition that receives the energy (cyano- mode).19 Due to
the strong Fermi resonance, the energy flow from the azido-
to the dark transitions is much faster compared to that of
the dark transitions to the cyano- mode. As a result, a broad
negative peak appears indicating
relaxation into the cyano- mode. Over time, the positive
a delayed energy
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transition appears as the energy finally relaxes to final
mode.
PAMB has a weaker Fermi resonance slowing the
the mechanism involved. It should be noted that cross peaks
1
2
3
4
5
6
7
8
occasionally arise through thermal transfer which result in
frequency shifts indistinguishable from direct excitation of
the accepting mode.19 Still, this thermal response is typically
observed in larger systems, and thus it is unlikely for these
small model systems. However, we cannot completely rule
out some contribution from this effect.
energy transition from the azido- mode to the dark states.
The coupling strength between the cyano- and combination
bands are similar for both PAB and PAMB where the
anharmonicity of the cross peaks are 12 cm-1. Thus, the IVR
transfer from the azido- to the cyano- mode in PAMB is
slower, allowing the positive-going cross peak not to be
masked as discussed above for PAB. An oscillatory feature
is observed in the cross peak decay. It is likely due to
vibrational beating between transitions with an oscillating
frequency of ~13 cm-1.
One approach to quantify the rate of energy transfer is
to measure where the cross peak intensity reaches a
maximum, as demonstrated by the Rubtsov group.19, 48, 53, 62-
9
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63, 65, 81
.
Figure 4 shows the growth and decay of the positive
cross peak intensities for both PAB and PAMB. In this model
the time it takes for the cross peak intensity to reach a
maximum is the vibrational population transport time from
the azido- to the cyano-. The decay time represents cooling
of the cyano- mode.81 For PAMB, the cross peak intensity
begins to increase at T=800 fs and exhibits a relatively
smooth growth and decay. For PAB, the cross peak appears
much later at T=1.6 ps. As mentioned earlier, the negative
cross peak is broad for PAB at early waiting times and
overlaps with the positive peak interfering with the
intensity. Thus, the positive cross peak growth and decay
exhibits more noise. Initially, both intensity profiles were
fitted with an unconstrained bi-exponential growth and
decay function to measure the time of maximal intensity.
The original growth rate for PAMB is half of PAB suggesting
62
multiple pathways for IVR. The decay rates were similar
Figure 4. 2D IR cross peaks 5 and 5’ of Figure 1 intensity
growth and decay for PAMB (red) and PAB (blue). The data
was fit to a bi-exponential growth and decay.
which is expected due to the similar vibrational lifetimes of
the cyano- transition. The data were then fitted with a
constrained bi-exponential, which is shown in Figure 4,
where the decay rate was constrained to the vibrational
lifetime of the cyano-. The growth rate of PAMB was still
lower than that of PAB.
Cross Peaks Dynamics
At the intersection of the azido- and cyano- transitions,
cross peaks appear in the off diagonal at later waiting times,
T>0 as a result of IVR between the transitions. Peaks 5 and
5’ in Figure 1 show both a growth and a decay. Since dipolar
coupling is detected in PAMB within the same spectral
region as the IVR, it can influence the growth and decay rate
of the cross peaks during the IVR process. However, with
such a short lifetime (~300 fs), the contributions of dipolar
coupling are essentially gone (~5% at 800 fs) before the
intensity of the cross peaks starts to increase (see SI). Thus,
in this case, the influence is not significant.
Intramolecular vibrational energy redistribution
between vibrational modes occurs in polyatomic molecules
by energy transport from one mode to the others via
multiple anharmonically coupled low frequency modes.
Knowing the pathway for energy transfer is difficult as the
coupling is not just between the two modes of interest but
instead it involves several combination bands and other
dark modes that are not easily detected experimentally.
However, analyzing the rate of energy transfer and
comparing these results to theory can aid in the
identification of such modes and create new insights into
As shown in the graph, PAMB reaches a maximum prior
to PAB which does not initially make sense as the pathway
from the azido- to the cyano- is shorter. However, while a
correlation between bond distance and time of maximum
spectral amplitude have been covered in previous
experiments,48, 53, 63 the structure of the molecule can also
influence the rate as well, along with bond distance which
has been shown in other benzene systems.62 The structural
flexibility influences the overlap between the modes of the
system thus influencing the rate of energy transfer. In our
case the methylene group creates a difference between the
two molecules where the azido- in PAMB can exhibit more
movement (such as a rotation or wiggle). Also, the negative
band present at early waiting times for PAB does influence
the amplitude of the growth function. In the bi-exponential
fit for PAMB the amplitudes are essentially equal where at
T=0 the cross peak intensity is approximately zero which is
expected. When the amplitude of the growth function is set
equal to the amplitude of the decay constant the Tmax was
recalculated for PAB yielding a value of Tmaz=3.1 ps. The
corresponding transport rate is 227 m/s. For PAMB the
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Tmax was determined to be 4.6 ps, resulting in a transport
Page 8 of 21
푋푌푁30
푋 ― 푌
― 푒휆 푡 + 푒휆 푡
]
1
2
[
퐶푃 =
(4)
1
2
19
speed of 186 m/s (see SI for details).
Equation 3 was used for the intensity of cross peak 5’ of the
PAB compound, while equation 4 was used for the fit of the
intensity cross peak 5 for the PAMB compound. The
amplitudes for the growth and the decay of the kinetic trace
should be equal since the energy transfer into the mode
should be the same as the energy lost as the mode cools.
However, for PAB they are not. Thus, a weighting factor of
3.5 in equation 3 was added to account for the difference in
amplitudes in the growth and decay exponential rates
3
4
5
6
7
8
9
Kinetics
Although using the cross peak intensity maximum has
been shown as an effective method to measure rates of
energy transfer, another approach presented in the
literature is a simple kinetic model which relates the
vibrational lifetimes with the rates of energy transfer from
one transition to another.21 In our case this model can be
represented as the following scheme:
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푁
discussed above (see SI for details).
is the population of
30
the azido- mode after the initial equilibrium. To account for
any loss of population due to the initial fast equilibrium
during the vibrational relaxation, a factor (29% for PAB and
15% for PAMB) was multiplied by the amplitude from the
slow component in the bi-exponential decay. In equation 3
and 4
푘푁
퐾푒푞
3푁3 퐶푁푘퐶푁
푘
푘퐶푁
are the experimentally measured
where
and
푁3
vibrational decay constants of azido- and cyano-,
respectively.
푘
is the energy transfer rate (as indicated
퐶푁→푁3
by the cross peaks 7 and 7’ in Figure 1) from the cyano- to
azido- mode and 푁 →퐶푁 is the energy transfer rate from the
퐾1 ― 퐾2 2 + 4
(
푘푁3→퐶푁)(푘퐶푁→푁 )
3
(
)
(
)
퐾1 + 퐾2 +
푘
3
휆1 =
휆2 =
,
2
azido- to the cyano- mode. In both cases, the azido- lifetime
was bi-exponential in nature with a slow component and a
fast one. The fast component is not used in the analysis
because of the time scale of the cross peak dynamics. The
cross peaks arise around T≈1 ps and the fast lifetime
component is much faster, T<1 ps. The fast component
simply allows the fundamental and dark modes to reach an
equilibrium and reduces the population of the azido- mode
(down to 29% for PAB and 15% for PAMB), thus the
population in the kinetic model was adjusted
appropriately.80 The long component is the rate
determining step in the IVR process and used in the kinetic
scheme.
퐾1 ― 퐾2 2 + 4
(
푘푁3→퐶푁)(푘퐶푁→푁 )
3
(
)
(
)
퐾1 + 퐾2 ―
,
2
퐾1 ― 퐾2 2 + 4
(
푘푁3→퐶푁)(푘퐶푁→푁 )
3
(
(
)
(
(
)
― 퐾1 + 퐾2
+
푋 =
푌 =
,
2푘퐶푁→푁
3
퐾1 ― 퐾2 2 + 4
(푘푁3→퐶푁)(푘퐶푁→푁 )
)
)
― 퐾1 + 퐾2
―
3
퐾1 = ― 푘푁
, and
A
3
2푘퐶푁→푁
3
― 푘푁 →퐶푁
퐾2 = ―푘 ― 푘퐶푁→푁
.
퐶푁
3
and
fit to the
3
experimental data was achieved by variation of only the
푘
parameter. From the least squares fit, the azido- to
퐶푁→푁3
-1
푘
= 0.540 ± 0.151
휏
(
푁3→퐶푁
cyano- transfer rate is
ps
푁3→퐶푁
Since the vibrational lifetimes are directly measured
experimentally, the energy transfer rates,
ps-
푁 →퐶푁 = 0.0434 ± 0.0109
3
푘
=1.9±0.532 ps) for PAB, and
푘
and
푁3→퐶푁
1
휏
(
푁 →퐶푁=23.0±5.75 ps) for PAMB. As expected, the transfer
3
푘
, are determined from the fit of the kinetic model.
퐶푁→푁3
rate is faster for PAB compared to PAMB (See SI for back
transfer data and rates).
The rate constants are related by detailed balance:
푘푁 →퐶푁
퐸퐶푁 ― 퐸푁
3
3
퐾푒푞
=
= exp
―
(
)
To compare these results to our theoretical
studies, the IVR rate was determined by adding the
calculated IVR rates for each of the components along the
proposed pathway; i.e. the IVR rate from the azido- to the
Fermi resonance and the combination bands within the ring
to cyano- (via microscopic reversibility) with the strongest
coupling. While multiple possible pathways likely exist for
IVR, only the most probable ones are considered with the
highest TFR value. In PAB, the IVR rate from the azido- to
푘퐶푁→푁
푅푇
3
퐸
퐸
푁3
where R is the Boltzmann constant,
and
are the
퐶푁
respective excitation energies, and the temperature is 293
K (approximately the room temperature). With 푁 →퐶푁 and
푘
3
푘
as the fitting variables, the principle of detail balance
퐶푁→푁3
푘
= 푘푒푞 ∗ 푘
퐶푁→푁3
allows for substitution of
. The details
푁3→퐶푁
of the model are elaborated in the SI.
A least-squares fit was performed for the growth and
decay of the cross peak intensity (CP) of 5 and 5’ in Figure 1
and is shown in Figure 5. The following equations were
utilized to obtain the IVR rate from the azido- to the cyano-
the combination band is 푁 →퐹푒푟푚푖=2.6 ps-1
(
푘
휏
푁 →퐹푒푟푚푖=0.38
3
3
ps) and the IVR rate from the combination band to the
cyano- is 푅푖푛푔→퐶푁=0.068 ps-1
(
푘
휏
푅푖푛푔→퐶푁=14.8 ps) which is
mode
much slower due to being delocalized. Thus, the total
energy transfer rate is determined by the following
equation:
푋푌푁30
―푒휆 푡 + 3.5 ∗ 푒휆 푡
]
1
2
[
퐶푃 =
(3)
푋 ― 푌
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artificial measure of the rate constant and deviates from the
1
1
1
=
+
1
2
3
4
5
푘
푘
푘푅푖푛푔→퐶푁
actual IVR growth rate. Despite the difference from the
theoretical values, a singular pathway can be identified due
to the large TFRαβγ (> 1.9) for only one combination band. In
PAMB, with less congestion is observed in the cross peak
region of the 2D IR spectrum. Thus, the positive-going cross
peak is always clearly present and exhibit a smooth growth
and decay of similar amplitudes, unlike the PAB.
푁3→퐶푁
푁3→퐹푒푟푚푖
-1
푘
= 0.066
휏
( 푁 →퐶푁=15.1
3
yielding a transfer rate of
ps).
ps
푁3→퐶푁
As for the PAMB compound, the theoretical energy
transfer rate from the azido- to the cyano- mode was
calculated in a similar fashion. However, as discussed
earlier, there are two highly probable pathways from the
, 훷
,
αβγ
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7
8
9
Overall, the strength of the Fermi resonance depends
on the energetic degeneracy of the fundamental and
combination bands that are anharmonically coupled.
Stronger coupling between modes results in a pathway for
vibrational energy to propagate within the molecule. Both
our experimental and theoretical results indicate that a
stronger Fermi resonance is present in PAB, leading to a
faster IVR rate. Although PAMB also exhibits some
anharmonic coupling as shown in both the FTIR and 2D IR,
the IVR rate of transfer is significantly slower. Due to the
lack of strong coupling in PAMB, the energy transfer
pathway requires multiple pathways (at least two) from the
azido- mode to propagate through the molecule to the
cyano- mode.
azido- to the cyano- based on the values of the TFRαβγ
.
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훥휔
and
. The figure of merit for strongly anharmonically
coupled modes is the TFRαβγ as it considers all the relevant
parameters. The IVR rate from the ring to the cyano- mode
is similar to the PAB pathway, having a value of
=0.09 ps-1
energy to flow from the azido- to the ring have IVR rates of
푘
ps-1
푘푅푖푛푔→퐶푁
휏
(
푅푖푛푔→퐶푁=11.1 ps). The two possibilities for the
=0.26 ps-1
(
=0.05
휏
푘′
=3.9 ps) and
푁3→퐹푒푟푚푖
푁3→퐹푒푟푚푖
푁3→퐹푒푟푚푖
휏′
(
푁 →퐹푒푟푚푖=18.3 ps). Since both of these combination
3
bands have a TFRαβγ > 0.7 (See Table 1), each mode was
analysed as a part of the primary pathway of energy
transfer giving a range of IVR times from 15 to 33 ps. The
experimental value falls well within this range suggesting
that the major contributions to the pathway have been
identified. To further refine the model, an approach to
include both competing pathways was instituted to acquire
a theoretical rate. In this approach, the contribution of each
competing pathway was weighted by the calculated TFR
values, since these values represent the strength of the
anharmonic coupling between the azido- and the
combination bands. Since the combined total TFR value for
two pathways is 1.716 (a similar value to PAB), the
fractional contribution of each pathway would be 0.45 and
0.55 for the TFRαβγ values of 0.77 and 0.94, respectively.
With these weighted contributions, the IVR time for the
휏
combined pathway would be 푁 →퐹푒푟푚푖=0.45*3.9 ps +
3
0.55*18.3 ps = 11.8 ps. With this value, the total transport
휏 = 휏
+ 휏′
+ 휏푅푖푛푔→퐶푁
,
푁3→퐹푒푟푚푖
time,
is 22.9 ps.
푁3→퐹푒푟푚푖
Through this combined transfer pathway model, the IVR
process matches well with experiment and thus the
pathway is identified.
The overall trend in the theoretical rate constants
matches the experimental results, i.e. PAB is faster.
Furthermore, the magnitude of the experimental and
theoretical constants for the PAMB is within < 1% showing
that our simplistic approach captures the most important
contributions of the energy transfer pathway. Due to
significant congestion in the cross peak region of the
experiment, the theoretical and experimental rates show a
much larger deviation (~7-fold) for the PAB compound. In
particular, the major overlap occurs when the negative-
going cross peak, present at T=0, cancels out the positive-
going cross peak until much later waiting times. As a result
of this spectral overlap, the initial slope of the PAB kinetic
trace rises more rapidly than PAMB. Thus, it creates an
Figure 5. Growth and decay of cross peak (top) 5’ and (bottom)
5 from Figure 1 for PAB and PAMB compounds, respectively.
The solid lines represent the fit to the derived kinetic model.
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Page 10 of 21
By reducing the coupling strength via the addition of a
single carbon atom between the ring and the azido- group,
the resulting bottleneck effect not only decreases the rate of
IVR but still maintains slightly favorable energy flow in the
forward direction. It should be noted that tuning the Fermi
resonance strength, the propensity of energy to flow in one
direction can become either less or more favorable.
Developing methods to modify energy transfer in this way
will allow reactivity to be guided for desired chemical
In addition to varying the rate of transfer, the
1
2
3
4
5
6
7
8
direction of the IVR pathway from the azido- to cyano-
can be somewhat regulated via the presence of a strong
Fermi resonance. The Fermi resonance with the azido-
group formally locks in the fast transfer on one end of
the molecular scaffold, while the lack of the Fermi
resonance slightly inhibits back flow on the other side.
Even in the PAMB compound where two pathways are
necessary, the presence of the weaker Fermi resonance
is enough to cause the pathway to remain slightly
favored. As for comparison to the literature, the correlation
between the transport time and distance uncovered by
Rubtsov62 was well within the typical range found for IVR
in other scaffolds (150 to 500 m/s). Modifying the Fermi
resonance alters the energy flow from one transition to
another. These subtle changes can translate to changes
in chemical synthesis. Varying the energy dissipation
within the molecular system can improve/reduce the
required energy of a reactive pathway leading to
9
reactions and product formation.
During chemical
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reactions, energy is transported via vibrations from one
nucleus to another. Modifying this energy flow can lead to
more efficient chemical synthesis. This work and previous
literature show that the rate of energy transfer can be tuned
by the strength of the Fermi resonance.44-45
Also, the IVR studied here is relevant in the design of
energy diodes.30, 39 In particular, our studies demonstrate
that vibrational energy rectification can be exploited for
molecular electronics.
selectivity.
Recently, for example, Scholes and
coworkers found that a terpyridine-molybdenum
complex containing a dinitrogen bridge exhibits a
Fermi resonance that can be activated by light to
transfer energy to the spatially separated dinitrogen,
facilitating reactions involving N2.45
Conclusions
Intermolecular vibrational energy redistribution has
the power to influence the underlying fundamental
principles of chemistry ranging from chemical reactivity to
energy flow in diodes. In this manuscript, we have
described the details of the coupling processes for these
systems and demonstrated that molecular energy flow is
tunable via alteration of Fermi resonances and the intrinsic
coupling manifold of the molecule. Serving as a model
system, this approach for uncovering the coupling manifold
between these two transitions expands the notion that the
transfer pathway can be influenced to guide the energy flow
and rectification, which is necessary for design of molecular
electronics. In addition, with a better understanding of the
mechanisms at work, the IVR pathway can be limited to one
favorable direction making molecular selectivity in
synthesis and biological function become possible.
In this work, the energy transfer pathways and
rates were determined for both PAB and PAMB
compounds. With the similarity in scaffold, the rate was
modulated by variation of the coupling present along
the transfer pathway. Thus, the transfer time in PAB
(~2.0 ps) was slowed by more than an order of
magnitude via only a slight change in the coupling, i.e.
localization of the azido group. For PAMB (IVR~23.0
ps), the methylene group decoupled the azido- mode
from the ring to modulate the strength of the Fermi
resonance that guides the energy transfer pathway.
These results are substantiated by the theoretical
calculations indicating a decrease in the TFR and by the
experimental results showing a decrease of Fermi
coupling peaks in the 2D IR spectrum. Our results
provide a systematic roadmap to vary energy transfer
within an aromatic scaffold by modulating the
associated Fermi resonances, showing that the simple
addition of a methylene group can slow down energy
transfer rate by >10 fold.
Using the approach put forth in this manuscript, the
possible transport pathways can be systematically
evaluated for their contributions to the overall transfer
rate. During the analysis, the experimental rates are
compared to the theoretical rates determined from
each possible pathway, guided by the third order
resonance factor, TFRαβγ. Ultimately, the closest match
to the experimental value of the transfer rate reveals
the major contributing pathway. In the PAMB model
system, the theoretical rate constant matches almost
exactly the experimental values, showing the strength
of this combined approach. Unfortunately, for the PAB
model system, although the pathway is easier to
identify from theory, the approach of matching the rate
constants is complicated by the congestion in the cross
peak region, leading to less accuracy in matching the
experimental values. Overall, our work shows that with
a combination of experiment and theoretical analysis,
the IVR pathway can be determined and the rate
subsequently tuned by modulation of the coupling
manifold within molecular systems.
ASSOCIATED CONTENT
Details of the synthesis, theoretical data, the cyano- to the
azido- cross peak analysis, analysis on the bi-exponential fit of
the cross peaks, derivation of the kinetic scheme, and dipolar
coupling can be obtained in the Supporting Information. This
material is available free of charge via the Internet at
http://pubs.acs.org.
AUTHOR INFORMATION
Corresponding Author
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The Journal of Physical Chemistry
* Dr. Matthew J. Tucker, Email: mtucker@unr.edu
filter revealed by 2D IR spectroscopy. Science
2016, 353 (6303), 1040-1044.
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4
5
6
7
8
Author Contributions
7.
Anna, J. M.; Kubarych, K. J., Watching
MJT, DML, EEF and SHB designed the research. AJS and FC
performed the laser experiments. AJS and MJT provided
expertise in setup and optical design. HDP and DML calculated
the theoretical data. TS and EEF synthesized the model
systems; TS and SHB measured preliminary IR data; AJS, FC,
HDP, DML, and MJT analyzed the data; AJS, FC, HDP, EEF, DML,
and MJT wrote the paper. All authors revised and edited the
manuscript.
solvent friction impede ultrafast barrier crossings:
A direct test of Kramers theory. J. Chem.Phys.
2010, 133 (17).
8.
Bagchi, S.; Charnley, A. K.; Smith, A. B.;
Hochstrasser, R. M., Equilibrium Exchange
Processes of the Aqueous Tryptophan Dipeptide.
J. Phys. Chem. B 2009, 113 (24), 8412-8417.
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Funding Sources
9.
Kim, Y. S.; Hochstrasser, R. M., Chemical
The research was supported by NIH (R15GM1224597) to MJT,
NSF CHE-1361776 and CHE-1854271 to DML, Henry Dreyfus
Teacher-Scholar Award (TH-15-009) to SHB, and NIH
(R15GM093330) to SHB/EEF.
exchange 2D IR of hydrogen-bond making and
breaking. Proc. Natl. Acad. Sci. U. S. A. 2005, 102
(32), 11185-11190.
10.
Asplund, M. C.; Zanni, M. T.;
ACKNOWLEDGMENT
Hochstrasser, R. M., Two-dimensional infrared
spectroscopy of peptides by phase-controlled
femtosecond vibrational photon echoes. Proc.
Natl. Acad. Sci. U. S. A. 2000, 97 (15), 8219-8224.
We thank Gwendolyn Fowler for the synthesis of PAMB.
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rotational isomerization in room-temperature
solution. Science 2006, 313 (5795), 1951-1955.
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