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2015, 17, 4739
Laser-induced fluorescence of free
diamondoid molecules
Robert Richter,*
a
Merle I. S. Ro
¨hr,
b
Tobias Zimmermann,
a
Jens Petersen,
b
Christoph Heidrich,
a
Ramon Rahner,
a
Thomas Mo
¨ller,
a
Jeremy E. Dahl,
c
Robert M. K. Carlson,
c
Roland Mitric,
b
Torbjo
¨rn Rander
a
and Andrea Merli
a
We observe the fluorescence of pristine diamondoids in the gas phase, excited using narrow band ultraviolet
laser light. The emission spectra show well-defined features, which can be attributed to transitions from the
excited electronic state into different vibrational modes of the electronic ground state. We assign the normal
modes responsible for the vibrational bands, and determine the geometry of the excited states. Calculations
indicate that for large diamondoids, the spectral bands do not result from progressions of single modes, but
rather from combination bands composed of a large number of Dv= 1 transitions. The vibrational modes
determining the spectral envelope can mainly be assigned to wagging and twisting modes of the surface
atoms. We conclude that our theoretical approach accurately describes the photophysics in diamondoids
and possibly other hydrocarbons in general.
1 Introduction
Since their successful isolation and purification from petroleum in
the early 2000s,
1
hydrogen-terminated nanometer-size diamonds
of various sizes and shapes (diamondoids), with a sp
3
hybridized
carbon framework, have received much attention. Besides their
unique properties,
2
which have been the subject of many experi-
mental and theoretical studies,
2–10
diamondoids serve as building
blocks for anti-viral agents and drugs for treatment of neuro-
degenerative diseases.
11
Their negative electron affinity makes
them potentially useful for electron-emitting devices.
12
Beginning
with tetramantane, the higher members of the series include
several structural isomers, which make diamondoids ideal candi-
dates for studying size and symmetry effects on spectroscopic
properties. As an example, in the case of pentamantane, the six
isomers vary from the highly symmetric T
d
-point group with
24 symmetry elements to the C
1
point group exhibiting only the
identity symmetry element. Furthermore, some higher isomers
also exhibit chirality.
In order to gain information on the geometric, electronic,
and optical properties of diamondoids, various spectroscopic
methods have been applied. X-ray absorption spectroscopy has
been employed to characterize the electronic structure of the
diamondoids, showing pronounced differences compared to
crystalline bulk diamond.
13
It has been found that, in the
one-electron picture, the energies of the lowest unoccupied
molecular orbitals (LUMOs) are almost independent of the
diamondoid’s size and are determined by the hydrogen surface
atoms.
13
By contrast, the energies of the diamondoids, HOMOs
exhibit clear size-dependent shifts caused by quantum confine-
ment effects.
14
Moreover, valence band absorption spectra show
a strong shape dependency of the optical properties.
15
The
ionization potentials (IPs) of the diamondoids from adamantane
to pentamantane were determined by photoion yield spectro-
scopy, indicating a monotonic decrease with increasing cluster
size.
16
Experimental and calculated Raman spectra of a large
variety of pristine diamondoids from adamantane to hepta-
mantane show that each diamondoid has a unique vibrational
structure.
17
Using optical rotatory dispersion and vibrational
circular dichroism spectroscopy, the chirality of [123]tetramantane
was observed.
18
A study of the structure of the adamantane cation
19
provides evidence of a Jahn–Teller distortion in the cationic
electronic ground state.
Size- and shape-selected neutral diamondoids in the gas
phase are optimal model systems for theory, especially for the
study of quantum confinement effects.
20–25
In particular, doped
or surface-modified diamondoids
2–6
as well as diamondoid
dimers or trimers are extremely attractive as nanoscale building
blocks for applications.
Another interesting feature is the photoluminescence of
diamondoids. The emission spectrum of adamantane in the
gas phase, recorded by photoexcitation with synchrotron radia-
tion, shows a broad, structureless intrinsic luminescence in the
a
Institut fu
¨r Optik und Atomare Physik, Technische Universita
¨t Berlin,
b
Institut fu
¨r Physikalische und Theoretische Chemie, Universita
¨tWu
¨rzburg,
Am Hubland, 97074 Wu
¨rzburg, Germany
c
Stanford Institute for Materials and Energy Sciences, Stanford University,
Stanford, California 94305, USA
Received 30th September 2014,
Accepted 30th December 2014
DOI: 10.1039/c4cp04423a
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UV spectral region,
26
the origin of which, however, could not
be clearly identified. The fluorescence spectra of crystalline
diamondoids are found to be structureless with maxima
around 295 nm, independent of their size and shape.
27
In a
recent study, we presented size- and shape-dependent photo-
luminescence and excited state lifetimes for several diamondoids
in the gas phase recorded using synchrotron radiation,
28
enabling
us to resolve vibrational structure for several diamondoid mole-
cules. However, due to the comparatively low fluence of the
synchrotron light, excitation at the optical gaps was not always
possible and prevented a resolutionofthefinestructureinsome
cases. Quantum chemical calculations performed for adamantane
indicate geometrical changes of the molecular structure after
excitation. The photoluminescence of diamondoids is attributed
to vibronic transitions between vibrational manifolds of the
electronic ground and the first excited state. These initial
findings are confirmed also by calculations based on time-
dependent correlation functions performed by Banerjee et al.
29
who have reported theoretical vibrationally resolved absorption,
emission and Raman spectra of diamondoids.
In the present work, we present the first laser-excited fluores-
cence spectra of higher diamondoids (from triamantane to penta-
mantane) and compare them with theoretical vibrationally
resolved emission spectra obtained by quantum chemical calcula-
tions. The excitation source was an OPO laser system with a
narrow bandwidth, providing approximately three orders of mag-
nitude higher photon flux in the energy range below 6 eV than
synchrotron radiation. This, in combination with a high-resolution
fluorescence spectrometer, enabled us to resolve the fine structure
of the fluorescence and examine it as a function of the excitation
energy and laser fluence. Moreover, calculations based on the
Franck–Condon principle yielded vibrationally resolved emission
spectra that match the experimental data to an excellent degree.
This allowed for the analysis of the vibrational structure and the
assignment of the optically activenormalmodesaswellasforan
identification of size and symmetry effects in the spectra of
diamondoids. To our knowledge, this is the first systematical study
with an experimental and theoretical analysis of the emission
spectra of such complex molecular structures.
2 Experimental section
The perfectly size- and shape selected samples used in this
experiment originate from crude oil. Their purity (499%) was
determined by GC-MS analysis, and structures were confirmed
by single crystal X-ray diffraction.
1
A gas cell integrated in a
vacuum chamber was used for heating the samples to tempera-
tures at which they sublimed. The cell was evacuated to an
ambient pressure of 10
5
mbar before heating. UV-transparent
windows allowed the interaction of the laser light with the gas
phase diamondoids. A nanosecond Nd:YAG pumped optical
parametric oscillator (Continuum Powerlite 9010, Panther EX
OPO) with a UV-doubler extension was used for the excitation.
The laser system generated ns pulses at a 10 Hz repetition rate,
with a spectral width of 0.3 meV and pulse energies up to 80 mJ
at 2.75 eV. The pulse energy drops to about 1 mJ at 6.05 eV. The
laser beam was focused in the center of the gas cell. The
fluorescence light was detected perpendicularly to the laser
propagation using a 0.3 m Czerny–Turner spectrometer (Andor
Shamrock SR-303i) with a cooled CCD camera as detector.
The fluorescence light was focused on the entrance slit by
achromatic UV lenses. For all measurements, a grating with
1200 lines per mm and a 20 mm entrance slit was used with a
resulting spectral resolution of about 3.8 meV at a center energy
of 5.5 eV.
3 Computational methods
The vibrationally resolved optical emission spectra of the
diamondoids were calculated based on the Franck–Condon
principle
30,31
using the method of Santoro et al.
32,33
as imple-
mented in the Gaussian 09 quantum chemical software package.
34
This approach involves the calculation of harmonic vibrational
normal modes for the optimized geometries of the ground and
excited electronic states contributing to the vibronic band of
interest, followed by the computation of Franck–Condon factors
for the individual combinations of vibrational states. For this
purpose, the Duschinsky transformation between the normal
modes of the ground and excited electronic state was performed,
35
and the Franck–Condon factors were obtained analytically,
utilizing a recursive procedure.
36
The calculation of the emission
spectrum of [121]tetramantane, which has a dark first excited
state, was performed using the Herzberg–Teller approximation
43
employing the FCClasses program of Santoro et al.
32,37–39
This
involves a linear expansion of the transition dipole moment in
terms of the molecular coordinates and thus goes beyond the
standard Franck–Condon approximation of a constant transition
dipole, which would predict no emission intensity in this case.
In the present calculations, only transitions from the vibrational
ground state of the initial electronic state to the manifold of
vibrational states of the final electronic state were taken into
account. Notice, that nonadiabatic coupling effects between the
vibrational manifolds of different electronic states have not been
included. In the experiment, the laser excitation was performed at
the onset of the absorption band, thus only the lowest vibrational
states of the first excited electronic state are populated. Since these
are energetically well separated from the vibrational manifolds
of higher electronic states, nonadiabatic effects should be
insignificant. The electronic structure was described using density
functional theory (DFT) for the electronic ground state and time-
dependent density functional theory (TDDFT) for the excited states.
Due to the Rydberg character of the diamondoids’ low-lying
electronically excited states, the long-range-corrected CAM-B3LYP
functional
40
combined with the 6-31++G basis set
41
containing
diffuse basis functions was employed. This ensures an accurate
description of these states, as is evident from the very good
agreement between the theoretical and experimental spectra. The
calculated stick absorption and emission spectra were convolved
with a Gaussian function in order to obtain better comparability
with the experimental data.
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4 Results and discussion
We will first show the measured spectra one by one and compare
them to their calculated counterparts. After the spectra have been
analyzed individually, we will discuss how the composition of the
spectra changes with the size of the diamondoids. Finally, we will
examine which types of modes are ‘bright’ (lines with significant
oscillator strength contributing to the spectra) and provide a link
of these results to structural traits of the diamondoids.
4.1 Overview
In this work, the intrinsic luminescence of all investigated
diamondoids was recorded by exciting the samples close to
their optical gap energies. For several higher diamondoids,
absorption spectra and optical gap energies were published
by Landt et al.
15
and serve as a reference for this study. Since
the highest attainable photon energy of the used laser system is
E6.05 eV, the examined samples are restricted to diamondoids
with excitation energies in this range and lower (triamantane
and its higher homologues). Since the laser provides much
higher fluence than the synchrotron radiation used in previous
work,
28
the luminescence spectra could be recorded with much
higher resolution and lower excitation energies. This allows a
detailed comparison with theory and the accurate vibrational
analysis of the spectra.
Emission spectra of triamantane (C
18
H
24
,C
2v
-ground state
symmetry) are shown in Fig. 1 for five different photoexcitation
energies selected near the absorption onset at 6.06 eV.
15
The
strongest peak visible in each spectrum at high energies arises
from scattered laser light. The excitation peak is followed by the
fluorescence of the sample and extends about 1 eV to lower
energies. The fluorescence intensity depends on the energy of
the exciting photons and scales well with previously determined
absorption cross-sections.
15
Moreover, the emission spectra
exhibit fine structure, in the form of almost equally spaced
intense bands. A linear dependence of the entire fluorescence
signal on the incident laser intensity was observed up to 0.3 mJ
pulse energy. Thus, we conclude that the appearance of intrinsic
luminescence is the result of a one-photon excitation process.
Increasing the pulse energy above 0.3 mJ leads to the saturation
of the photoluminescence signal and causes the production of
free CH and C
2
radicals in the absorption cell. The free radicals
observed at higher laser intensities were identified by their
typical blue-green Swan bands in the emission spectra
42
and
are well known as laser evaporation products of carbon materials.
Further analysis of the photo-products, however, is not the
subject of this paper. In order to prevent fragmentation, the
incident pulse energy was kept around 0.1 mJ throughout
the experiments. A comparison with absorption spectra recorded
using synchrotron radiation shows an overlap between the lowest
energy peak in the absorption spectrum and the highest energy
peak in the emission spectrum of triamantane. This implies that
the same excited electronic state is involved in absorption and
emission. In most cases, this state corresponds to the lowest
excited state, which bears 3s-Rydberg character and involves
the HOMO–LUMO molecular orbital excitation.
24
The case of
[121]tetramantane, which has an optically dark first excited
electronic state, is significantly different and will be discussed
separately. The shapes of the emission spectra are determined
by the vibrational levels associated with the final state, in this
case the electronic ground state S
0
. The energetic position of
the emission spectra shown in Fig. 1 is almost independent of
the excitation energy. This clearly indicates that, regardless
of the excitation energy, the fluorescence arises from the same
initial quantum state, which is expected to be the ground
vibrational level of the first excited electronic state. In order to
verify this, the vibrationally resolved emission spectrum shown
in Fig. 2 was calculated for triamantane. This spectrum repro-
duces the experimental one very well. Moreover, the analysis of
the individual contributions giving rise to the overall spectral
envelope provides a deeper insight in the fine structure of the
spectra, which shows contributions of a large number of lines
with low intensity, associated to superpositions of several vibra-
tional normal modes.
The first peak (at E6.06 eV) in the calculated emission
spectra corresponds to the v0=0-v00 = 0 vibrational transition
between the S
1
and S
0
electronic states. In the case of triamantane,
this transition occurs at slightly higher energies than the photon
energy available from laser excitation. The absorption onset of
triamantane has the value of 6.06 eV (ref. 15), and the experi-
mental spectrum in Fig. 2 was recorded with an excitation
energy of 6.00 eV. The peak observed with an energy above the
laser excitation in the measured spectra results mainly from the
v0= 0 state that was initially excited from thermally populated
vibrational states (v00 40) in S
0
. A weak fluorescence intensity
from hot bands can also be observed for triamantane as well as the
other higher diamondoids. The analysis shows that the emission
spectrum of triamantane is mainly dominated by modes in the
frequency region 1100–1400 cm
1
(roughly 0.14 eV to 0.17 eV).
Fig. 1 Vibrationally resolved emission spectra of triamantane excited at five
different photon energies. The photoluminescence exhibits sharp features,
whose spectral positions are independent of the excitation energy. Each
spectrum is accumulated over 2 200 s, corresponding to 4000 laser
pulses. The laser pulse energy (0.15 mJ) and the sample temperature of 95 1C
were kept constant.
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These frequencies are associated to CH wag and CH
2
twisting
modes. The frequencies closely match the results of the Raman
measurements reported by Filik et al.
17
Previous calculations
for adamantane have shown that the tetrahedral symmetry
exhibited in the electronic ground state is reduced in the electro-
nically excited state.
28
Similarly, for triamantane, a change in the
molecular geometry after excitation is observed, and the calcu-
lated excited state structure shows no symmetry.
The addition of an isobutyl unit to different positions
on triamantane leads to the tetramantane (C
22
H
28
) isomers
[121]tetramantane with C
2h
-symmetry, [123]tetramantane with
C
2
-symmetry and [1(2)3]tetramantane with C
3v
-symmetry. Their
fluorescence spectra, recorded with the same excitation energy,
are shown in Fig. 3 and exhibit clear differences between the
isomers. Whereas [123]tetramantane and [1(2)3]tetramantane
show an intense fluorescence with clear fine structure, the
emission of the rod-shaped [121]tetramantane is characterized
by low intensity and only weak vibrational structures under the
same experimental conditions. The onset absorption energies
for [123]tetramantane and [1(2)3]tetramantane at 5.95 eV
and 5.94 eV, respectively,
15
lie in the range of the available laser
system. In [121]tetramantane, the lowest transition is dipole
forbidden, therefore a higher-lying 3p-like Rydberg transition
occurs.
15,24
The absorption spectrum thus shows a smooth onset
with the optical gap value of 6.10 eV.
15
This excitation energy is
not available with the present laser system, nonetheless a low-
intensity emission spectrum was observed by exciting the system
with a photon energy of 6.04 eV. This is due to the fact that
states that are dipole forbidden within the Franck–Condon
approximation may be accessed through Herzberg–Teller
43
vibronic
coupling given that the transition dipole moment changes upon
excitation.
Vibronic coupling can also be the reason for the lack of
structure in the measured luminescence spectra of [121]tetra-
mantane. Moreover, with increasing size of the diamondoid,
the temperature at which a sufficient vapor pressure is reached
in the cell rises, and thermal broadening effects tend to wash
out the vibronic fine structure of the measured spectra. This
makes it harder to compare the experimental spectra to their
theoretical counterparts than in the case of triamantane measured
at lower temperatures. The calculated vibrationally resolved emis-
sion spectra of [1(2)3]tetramantane and [121]tetramantane are
depicted together with the corresponding experimental spectra
in Fig. 4. To account for the different behaviour of [121]tetra-
mantane, compared to the other isomers, the spectrum of the
former has been calculated including Herzberg–Teller correc-
tions. In order to fit the measurement, the calculated spectra
were red shifted. Gaussian functions of different widths were
used to convolute the calculated stick emission spectra. The 0–0
transition between the S
1
and S
0
states for [1(2)3]tetramantane
at 5.98 eV is easy to assign. In the case of [121]tetramantane the
lack of structure in the experiment makes an assignment of the
spectral features difficult. The 0–0 transition is expected to be
covered by the laser peak.
The calculated spectra differ mainly in the low energy region:
the CCC-bending type modes at low frequencies in the rod-shaped
tetramantane isomer are more pronounced than the similar
modes in the more compact tetramantane. Similar to triamantane,
Fig. 2 Comparison between measured (red line) and calculated (blue line
and black sticks) spectra of triamantane (upper panel). The calculated spectra
are red-shifted by 0.1 eV with respect to the experimental spectrum in order
to fit the measurement. The blue line is the convolution of the stick spectrum
with a Gaussian function (FWHM 20 meV). The lower panel shows a
magnification of one band in the stick spectrum, emphasizing the large
number of vibrational normal modes of the electronic ground state contri-
buting to the spectral envelope.
Fig. 3 Emission spectra of the tetramantane isomers excited at 6.04 eV
and with pulse energies of 0.1 mJ. An isomer-dependent fluorescence
envelope is observed.
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several modes around 0.17 eV corresponding to CH wag/CH
2
twist vibrations contribute to the spectra of tetramantane.
The emission spectra of five different pentamantane iso-
mers excited at 6.04 eV are shown in Fig. 5. The calculated
spectra fit well with the experimental data for each investigated
isomer. The highest energy peak of the fluorescence spectra of
pentamantane is located between 5.83 eV and 5.92 eV.
The emission spectra of [12(1)3]pentamantane and [1213]penta-
mantane look very similar (Fig. 5c and d). Different modes
corresponding to combinations of CCC bend, CC stretch,
CH wag and CH
2
twist vibrations govern their fluorescence
envelope. In the case of the compactly shaped [1(2,3)4]penta-
mantane (Fig. 5e), the T
d
ground state symmetry is reduced to
C
3v
after excitation. The overall agreement between calculation
and experiment is, considering the size of the systems, remark-
ably good. The remaining differences are readily explained
mainly by thermal effects and the neglection of Herzberg–Teller
coupling. Studies on slightly smaller molecules have shown
that an inclusion of thermal effects results in the appearance of
tens of additional lines for each vibrational state.
39
Since our
spectra – calculated at 0 K – are already composed of several
thousand lines it becomes obvious that the diamondoids in
reality have very complex spectra. Taking thermal effects and
Herzberg–Teller coupling into account would lead to an inordinate
increase in the number of lines contributing to the spectra. With
rotational constants (%
B)aslowas0.2GHz(E10
6
eV) the
diamondoids also have a highly complex rotational structure.
At room temperature each vibrational line has an extremely
dense rotational fine structure with yet another several thousands
of lines of non-vanishing intensity kBT
hc
B
3104
.Considering
this large number of dense lines, a quick estimation shows
that the natural line width (E10
6
eV) is already one of the
limiting factors prohibiting a resolution of rotational structure.
Therefore, the width of a given vibrational line depends on the
temperature.
4.2 Size effects
At first glance, the luminescence spectra of all diamondoids up
to the pentamantanes look fairly similar. They all show a
distinct, rather broad fine structure with a Stokes shift of about
0.3 eV to 0.5 eV dominated by what seems to be progressions of
a few fundamental vibrational normal modes, forming bands
with an energetic spacing of about 0.17 eV. This band spacing is
almost the same for all of the diamondoids. Usually the Stokes
shift (i.e. the energetic difference between the 0–0-transition
and the band with the highest intensity) provides information
on the geometric distortion of the excited state with respect
to the ground state. When a molecule undergoes a strong
deformation upon excitation, the equilibrium positions of the
potential energy surfaces (PES) of the normal modes change.
This generally reduces the Franck–Condon Factors (FCFs) of
the 0–0 transition and increases the FCFs of overtones resulting
in an energetic shift between absorbed and re-emitted light.
Generally, the higher the overtones visible in the spectra (more
bands) the stronger the shift of the PESs. Our calculations,
however, indicate that this simple picture can not be applied
without reservation to the larger diamondoids. Calculated
spectra from ada- up to pentamantane are shown in Fig. 6.
The calculated spectra can be separated into different classes
characterized by the number of distinct ground state modes
each spectral line is composed of. Below the full spectrum (top
of each box) the individual classes are shown, beginning with
the class 0 consisting only of the 0–0 transition (v0=0-v00 = 0).
Class 1 is the simplest case of vibrational excitation, where each
line corresponds to one or more quanta excited in a single
normal mode (v0=0-v
x
00 =n
x
, with n=1,2,3,... and the
specific mode x). The emission lines of class 2, by contrast,
corresponds to the simultaneous excitation of two normal
modes (v0=0-((v
x
00 =n
x
)+(v
y
00 =n
y
))), with xay. Accordingly,
class 3 comprises the lines that are a combination of three
different fundamental normal modes (v0=0-((v
x
00 =n
x
)+
(v
y
00 =n
y
)+(v
z
00 =n
z
))), xayaz, and so forth. Analysing the
emission spectra in terms of these classes shows that, in the
lower diamondoids, the different bands visible in the spectra
are the result of progressions of a few intense normal modes up
to 5 overtones (nr6), as well as their combination bands.
Moving to the higher diamondoids this gradually changes.
Fig. 4 Emission spectra of [1(2)3]tetramantane (upper panel) and [121]-
tetramantane (lower panel). The calculated stick emission spectra were
convolved with Gaussian functions of different FWHMs, indicated in the
legend. Calculations are redshifted by 0.15 eV for [1(2)3]tetramantane and
0.3 eV for [121]tetramantane.
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In the case of the pentamantanes, the first band is the sum
of many lines (E20), all in the energy range between 0.14 to
0.176 eV, none of which shows significant overtones (nr1).
This means that the ‘higher’ bands observed in the spectra of
the higher diamondoids are almost exclusively combination
bands. To further elucidate this fact we investigate band 3 in
further detail. In Fig. 6, in the spectrum of adamantane, band 3
has contributions from class 1, 2, 3 and 4. In the case of
diamantane classes 1, 2 and 3 contribute. In tria- and tetra-
mantane mainly classes 2 and 3, while in pentamantane band 3
is almost solely composed of lines from class 3. Also, the other
bands in pentamantane exhibit a similar behavior. Band 1 are
the fundamental transitions (DvD1), band 2 are their combi-
nations in class 2 (2 DvD1) and band 3, as described, is the
combination of 3 fundamentals (3 DvD1), and so forth. This
also means that the average shift of the PESs is relatively small
for the higher diamondoids. The ‘higher’ bands in the spectra of
the higher diamondoids are the result of the large number of
modes, allowing for a large number of possible combinations in
the higher classes. That the ‘higher’ bands are well separated is
due to the fact that they are composed of fundamental normal
modes which stem from a rather small energetic interval.
These findings have further implications. In the smaller
diamondoids the vibrational progressions are fairly well resolved
even in the ‘higher’ bands, while the ‘higher’ bands of the larger
diamondoids are rather diffuse. This is readily explained by the
fact that these bands are a superposition of hundreds or even
thousands of lines with slight energetic differences, rather than
an envelope from a few normal modes with higher degrees
of excitation. In contrast to small molecules, the equi-energetic
spacing of the bands is almost independent of the actual shape
of the PES and does not indicate a harmonic behavior of the
potentials.
Another important aspect is that the ‘higher’ bands in the
larger diamondoids are comprised of an extremely large number
of lines. If, for convenience, one considers a case where all lines
in a given spectrum are 0–1 transitions in class 1 and combina-
tionsoftheseinthe‘higher’classes,thenumberofbrightlinesin
a class can be expressed by the binomial coefficient k
c
where kis
the number of bright 0–1 transitions and cis the class number.
Fig. 7 shows the number of possible lines for each class as a
function of ‘bright’ fundamental transitions. The pentamantanes
exhibit between 20 and 30 optically active modes in the first
band alone. Since for the pentamantanes each spectral band of
a given number (in descending energetic order) is dominated
by the combination class of the same number, the number of
lines in the n’th band can be also estimated by k
n
. This gives
rise to a very large number of possible vibrational lines, e.g. up
to 10
4
–10
5
in the 5th band. Although most of these lines only
have a small intensity, their large number adds up to signifi-
cant intensity even at high band numbers (low energies with
respect to the 0–0 transition).
An additional factor influencing the spectral shape is that
the spectral envelope is not only dependent on individual line
intensities but also on broadening effects. In spectra of small
molecules with few bright modes and no significant amount of
combination bands, the number of lines contributing to a given
band is almost independent on the band’s energetic position.
Fig. 5 Calculated (blue and green) and measured (red) emission spectra of five pentamantane isomers. Calculations are convolved with a Gaussian
function of different FWHM: blue lines FWHM of 20 meV, green lines: (b) and (d) FWHM of 100 meV, (c) and (e) FWHM of 124 meV. Calculations are
redshifted by (a): 0.19 eV, (b): 0.22 eV, (c): 0.28 eV, (d): 0.30 eV, (e): 0.14 eV.
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Therefore, the relative intensity of these bands does not signifi-
cantly change due to line broadening. The effective linewidth is a
combination of rotational population and experimental resolution
and is, in our case, on the order of the thermal energy, which
represents the largest contribution to the line broadening. In
contrast, in spectra in which each subsequent band is composed
of more lines than the preceding one, as in the case of the
pentamantanes, the ‘center of mass’ of the spectrum may shift as
the consequence of line broadening. A higher effective linewidth
should in principle emphasize bands with a higher number of
lines. This means that the experimentally derived Stokes shift
in these cases is also a function of the temperature and other
broadening effects. Since the temperature has only little impact on
the internuclear distances, it is apparent that the Stokes shift and
the positions of individual bands no longer have a clear relation to
a shift of the PES. As an illustration, Fig. 8 shows the calculated line
spectrum of [1(2,3)4]pentamantane. The spectrum has been con-
volved with Gaussian functions of different width. It becomes
evident that the effective linewidth has a strong influence on the
relative band intensities and the spectral shape drastically changes
with broadening. ‘Higher’ bands of pentamantane, which appear to
have negligible intensities in the line spectrum, grow substantially
when they are broadened by thermal effects.
4.3 Determination of line intensities
First of all, it has to be noted that symmetry considerations play
a key role in the selection of the optically active vibronic
transitions. Diamondoids with a larger number of symmetry
elements tend to have less optically active transitions, which in
turn, are on average a bit more intense. The main objective of
the following section, however, is to point out effects that are
independent of symmetry.
The 168 ground state normal modes of a pentamantane cover
a large energy range from 0.03 eV up to 0.38 eV. This raises
Fig. 6 Experimental (black) and calculated Franck–Condon emission spectra
(red) of adamantane, diamantane, triamantane (from top to bottom, left),
[1(2)3]tetramantane and [1(2,3)4]pentamantane (right) relative to the respective
0–0-transition energies. Below the full spectrum (top spectrum of each box)
the different classes (see text) composing the full spectrum are plotted.
Spectra of adamantane and diamantane are taken from ref. 28.
Fig. 7 Logarithmic plot of the number of lines in class 1 vs. number of
possible combinations in higher classes (C
2
,C
3
,...). These values corre-
spond to the number of possible lines in the higher’ bands of the larger
diamondoids. Due to the quantum nature of the vibrations, only integer
values are possible (dots), and the lines are guides to the eye.
Fig. 8 Calculated Franck–Condon spectrum of [1(2,3)4]pentamantane
(black, FWHM of 0.3 meV) and convoluted with different Gaussian functions
(FWHMs: green 13 meV, red 25 meV, blue 50 meV). All spectra are normalized
to their respective 0–0 transition.
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the question, why ‘bright’ modes only cover a very narrow range
from 0.14 eV to 0.18 eV. Answering question this would explain
the observation of the relatively well-defined bands seen in the
emission spectra.
To address this issue one has to take a closer look at the FCFs
which quantify the spatial overlap of initial and final vibrational
wavefunctions of a vibronic transition. The FCFs of a polyatomic
molecule are influenced by the transformation between the
normal modes of the initial and final electronic states, which is
determined by a shift vector and the Duschinsky rotation of the
normal modes. The shift vector is a measure for the displacement
of the PES minimum upon electronic excitation, while the
Duschinsky rotation matrix describes how the normal modes of
the final electronic state are expressed as linear combinations of
the modes of the initial electronic state.
37
Fig. 9 illustrates the structure of the Duschinsky rotation
matrices of [1(2,3)4]- and [1213]pentamantane together with
the respective shift vector for emission transitions. The two
pentamantane isomers have been chosen such as to represent a
highly symmetric ([1(2,3)4]pentamantane, T
d
symmetry) and a
nonsymmetric ([1213]pentamantane, C
1
symmetry) diamondoid.
In Fig. 9, dark colour denotes a high absolute value of the
matrix element, while light colour denotes a small value. Inside
the matrices, the spectral line intensities of the involved ground
state normal modes (red lines in Fig. 9) are plotted. Although
this is a rather unusual way of presenting spectra, it immediately
reveals to what extent a given normal mode contributes to the
spectrum. As the most obvious correlation, one finds that line
intensities scale to a good approximation with the absolute
value of the shift vector. This fact is easy to understand. When
the PESs involved in the transition have no shift with respect to
each other, the 0–0-transition is most probable. Thus, for a
different line to appear in the spectrum, a certain shift of the
two involved PESs is necessary. A closer look at Fig. 9 shows that
Fig. 9 Duschinsky matrices (values squared) of [1(2,3)4]- and [1213]pentamantane (top left, top right, respectively). The gray scale in the matrix reflects
the coupling of two modes. High values (E1.0) correspond to high coupling. The modes presented are in ascending order with respect to their energy.
Inside the matrix, the intensity of transitions from the vibrational ground state of the S
1
electronic state to the ground state modes is depicted (red, only
class 1 and no overtones). Panels bottom left and bottom right show the shift vectors (absolute value) of [1(2,3)4]- and [1213]pentamantane, respectively.
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not all ground state modes contributing to the spectra fulfill this
condition. In these cases, however, the mode expressed in the
spectrum shows strong mixing with another excited state mode
that exhibits a large shift vector.
In the following, it will be discussed why some types of
modes are ‘bright’, and a connection will be made from these
findings to the underlying structural traits of the diamondoids.
For this purpose, in Fig. 10, histograms of the bond lengths and
angles of [1(2,3)4]- and [1213]pentamantane in the ground and
the excited state are shown in order to visualize the structural
changes occurring upon electronic excitation.
The low frequency CCC-bending and the CC-stretching
modes generally have a small shift and very low amount of
mixing. The shift can be explained by the changing bond
lengths of the CC-bond between ground and excited state (see
Fig. 10), while the low amount of mixing can be attributed to
the fact that CCC-bending and the CC-stretching modes
generally involve the whole carbon framework (breathing-
like modes etc.). A rotation of these modes would imply a
very unlikely rotation of the whole carbon framework upon
excitation. Also, modes not involving all C-atoms are still
embedded in the relatively stiff carbon structure, hence rota-
tion of the PESs is hindered. The small shift and the negligible
mixing lead to an overall small contribution of these modes to
the spectra.
The CH wag and CH
2
twisting modes, by contrast, stand out,
since they generally have a large shift and show a significant
amount of mixing. The shift of these modes, however, cannot
be attributed to a change of the CH bond lengths upon
excitation. The direction of movement for these modes is
perpendicular to the direction of the bond, and the large shift
of the PES can almost entirely be attributed to a change of the
CCH angles upon excitation (see Fig. 10). Since the carbon
framework imposes very little restrictions on the wagging
plane, these modes have a high degree of freedom for rotation
and therefore show strong mixing. These factors make CH wag
and CH
2
twisting modes the dominant contributors to the
emission spectra.
The CH
2
scissoring modes neither shift nor mix, hence they
do not occur in the spectra. This could be attributed to a relatively
low change in the HCH angles upon excitation (see Fig. 10).
Furthermore the scissoring plane is always perpendicular to the
plane spanned by the CCC angle of the three nearest C-atoms.
Embedded in the stiff carbon framework and due to their high
mass being mostly unaffected by the scissoring motion, the
carbon atoms impose a direction of movement on the scissoring
modes. Consequently they show little mixing.
Finally, CH stretching modes exhibit almost no shifting but
a significant amount of mixing. The direction of the stretching
movement is always in the direction of the bond. The low shift
can therefore be explained by the low difference in the CH bond
lengths between ground and excited state. The relatively strong
change of the CCH angles after excitation allows for these
modes to mix. However, mixing occurs only with other CH
stretching modes, none of which has a significant shift, hence
these modes do not contribute to the spectra.
5 Conclusion
High-resolution laser-induced fluorescence spectra of diamondoids,
ranging from triamantane to pentamantanes and exhibiting a
variety of shapes and symmetries, have been investigated and
compared to theoretically calculated vibrationally resolved
emission spectra, computed within the harmonic approximation.
These are the first laser-induced emission spectra of gas phase
diamondoids. All studied systems exhibit characteristic emission
spectra with complex fine structure. Quantum chemical calcula-
tions were performed in order to provide insight into the details
of the distribution of vibrational states and on the overall shape
of the individual spectra.
The photoluminescence can be attributed to transitions from
the electronically excited state into a rich manifold of vibrational
modes in the electronic ground state. The calculated spectra
allowed for a detailed analysis of the vibrational structure of the
spectra and the assignment of the optically active normal modes.
The calculations show that the pronounced band structure
Fig. 10 (Upper panels) distribution of bond lengths of [1(2,3)4]- and [1213]-
pentamantane in the ground (S
0
) and excited states (S
1
). (lower panels)
Distribution of bond angles of [1(2,3)4]pentamantane (left) and [1213]-
pentamantane (right) in the ground (S
0
) and excited state (S
1
).
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observed in all emission spectra is, for the most part, not the
result of vibrational progressions but mainly due to combina-
tions of a large number of fundamental (0–1) transitions.
A significant part of these transitions originate from vibra-
tional modes associated with CH wagging and CH
2
twisting
modes on the diamondoid surface. This can be attributed to
a change of the equilibrium positions of the H-atoms in the
excited state, which can rather easily change positions com-
pared to the C-atoms in the stiff carbon framework. These
modes can be energetically combined to form higher bands
consisting of a large number of lines. Compared to lines from
higher overtones, these combination bands have, in principle,
no upper limit for the energy difference from the 0–0 transition,
while in overtone bands, the dissociation energy of the ground
state gives a natural upper limit for an energetic shift with
respect to the 0–0 transition.
Previous studies of diamondoid fluorescence have based
their conclusions on less well resolved experimental data
26,28,44
and could therefore reach only tentative conclusions. The much
increased quality of the spectra in the present work enables us
to corroborate some previous conclusions, such as the primary
dependence of spectral shape on particle symmetry, and almost
constant HOMO–LUMO gap between various sizes, whereas
some, such as the assignment of the broad emission spectral
shape to a self-trapped exciton, can be readily disproven. Thus,
the current study clearly illustrates the interdependence of a
detailed theoretical understanding of photophysical processes
in medium-sized to large molecules, and of high-quality experi-
mental data; it is clear that an intuitive interpretation of lower
quality spectra is insufficient.
Some recent studies have pointed out the necessity of taking
nuclear displacement into account when calculating spectra
of diamondoids, and have suggested a variety of methods to
do this.
25,29,45
In a previous study, we have already shown
that based on the calculation of vibrationally resolved Franck–
Condon spectra in the frame of the TDDFT method, which
includes the structural displacements upon changing the
electronic state, the experimental emission spectrum of ada-
mantane can be accurately reproduced.
28
Here, we have further
exploited this approach and have obtained a very good agree-
ment of the calculated vibrationally resolved spectra with their
measured high-resolution counterparts for a large class of
higher diamondoids. A thorough Franck–Condon analysis,
including Herzberg–Teller coupling if necessary, has provided
a detailed understanding of the form and composition of the
spectral features. We expect this to be the case also for other
diamondoids, diamondoid aggregates and large hydrocarbons
in general.
Acknowledgements
This work was supported by the Deutsche Forschungsge-
meinschaft DFG through funding of the research unit FOR
1282, grants FOR 1282 MO 719/10-1 and MI 1236/2 and by the
U.S. Dept. of Energy under contract DE-AC02-76SF00515.
References
1 J. E. Dahl, S. G. Liu and R. M. K. Carlson, Science, 2003, 299,
96–99.
2 H. Schwertfeger, A. A. Fokin and P. R. Schreiner, Angew.
Chem., Int. Ed., 2008, 47, 1022–1036.
3 G. A. Mansoori, P. L. B. De Araujo and E. S. De Araujo,
Diamondoid Molecules: With Applications in Biomedicine,
Materials Science, Nanotechnology and Petroleum Science,
World Scientific, 2012.
4 A. A. Fokin, B. A. Tkachenko, P. A. Gunchenko, D. V. Gusev
and P. R. Schreiner, Chem. – Eur. J., 2005, 11, 7091.
5 J. C. Garcia, J. F. Justo, W. V. M. Machado and L. V. C. Assali,
Phys. Rev. B: Condens. Matter Mater. Phys., 2009, 80, 125421.
6M.Vo
¨ro
¨s, T. Demje
´n, T. Szilva
´si and A. Gali, Phys. Rev. Lett.,
2012, 108, 267401.
7 R. Meinke, R. Richter, A. Merli, A. A. Fokin, T. V. Koso, V. N.
Rodionov, P. R. Schreiner, C. Thomsen and J. Maultzsch,
J. Chem. Phys., 2014, 140, 034309.
8 R. Meinke, R. Richter, T. Mo
¨ller, B. a. Tkachenko, P. R.
Schreiner, C. Thomsen and J. Maultzsch, J. Phys. B: At., Mol.
Opt. Phys., 2013, 46, 025101.
9 T. Zimmermann, R. Richter, A. Knecht, A. A. Fokin, T. V.
Koso, L. V. Chernish, P. A. Gunchenko, P. R. Schreiner,
T. Mo
¨ller and T. Rander, J. Chem. Phys., 2013, 139, 084310.
10 T. Rander, M. Staiger, R. Richter, T. Zimmermann, L. Landt,
D. Wolter, J. E. Dahl, R. M. K. Carlson, B. A. Tkachenko,
N. A. Fokina, P. R. Schreiner, T. Mo
¨ller and C. Bostedt,
J. Chem. Phys., 2013, 138, 024310.
11 W.J.Geldenhuys,S.F.Malan,J.R.Bloomquist,A.P.Marchand
andC.J.VanderSchyf,Med. Res. Rev., 2005, 25, 21–48.
12 W. L. Yang, J. D. Fabbri, T. M. Willey, J. R. I. Lee, J. E. Dahl,
R. M. K. Carlson, P. R. Schreiner, A. A. Fokin, B. A. Tkachenko,
N. A. Fokina, W. Meevasana, N. Manella, K. Tanaka, X. J. Zhou,
T. van Buuren, M. A. Kelly, Z. Hussain, N. A. Melosh and
Z.-X. Shen, Science, 2007, 316, 1460.
13 T. M. Willey, C. Bostedt, T. van Buuren, J. E. Dahl, S. G. Liu,
R. M. K. Carlson, L. J. Terminello and T. Mo
¨ller, Phys. Rev.
Lett., 2005, 95, 113401.
14 T. M. Willey, C. Bostedt, T. van Buuren, J. E. Dahl, S. G. Liu,
R. M. K. Carlson, R. W. Meulenberg, E. J. Nelson and
L. J. Terminello, Phys. Rev. B, 2006, 74, 205432.
15 L. Landt, K. Klu
¨nder, J. E. Dahl, S. G. Liu, R. M. K. Carlson,
T. Mo
¨llerand and C. Bostedt, Phys. Rev. Lett., 2009, 103,
047402.
16 K. Lenzke, L. Landt, M. Hoener, H. Thomas, J. E. Dahl,
S. G. Liu, R. M. K. Carlson, T. Mo
¨ller and C. Bostedt, J. Chem.
Phys., 2007, 127, 084320.
17 J. Filik, J. N. Harvey, N. L. Allan, P. W. May, J. E. P. Dahl,
S. Liu and R. M. K. Carlson, Spectrochim. Acta, Part A, 2006,
64, 681–692.
18 P. R. Schreiner, A. A. Fokin, H. P. Reisenauer, B. A. Tkachenko,
E. Vass, M. M. Olmstead, D. Bla
¨ser, R. Boese, J. E. P. Dahl and
R. M. K. Carlson, J. Am. Chem. Soc., 2009, 131, 11292.
19 A. Patzer, M. Schu
¨tz, T. Mo
¨ller and O. Dopfer, Angew. Chem.,
2012, 124, 5009–5013.
Paper PCCP
Published on 07 January 2015. Downloaded by TU Berlin - Universitaetsbibl on 15/06/2017 11:29:11.
View Article Online
This journal is ©the Owner Societies 2015 Phys. Chem. Chem. Phys., 2015, 17, 4739--4749 | 4749
20 G. C. McIntosh, M. Yoon, S. Berber and D. Toma
`nek,
Phys. Rev. B: Condens. Matter Mater. Phys., 2004, 70, 045401.
21 N. D. Drummond, A. J. Williamson, R. J. Needs and G. Galli,
Phys. Rev. Lett., 2005, 95, 096801.
22 A. J. Lu, B. C. Pan and J. G. Han, Phys. Rev. B: Condens.
Matter Mater. Phys., 2005, 72, 035447.
23 A. A. Fokin and P. R. Schreiner, Mol. Phys., 2009, 107, 823.
24 M. Vo
¨ro
¨s and A. Gali, Phys. Rev. B: Condens. Matter Mater.
Phys., 2009, 80, 161411.
25 C. E. Patrick and F. Giustino, Nat. Commun., 2013, 4, 2006.
26 L. Landt, W. Kielich, D. Wolter, M. Steiger, A. Ehresmann,
T. Mo
¨ller and C. Bostedt, Phys. Rev. B: Condens. Matter
Mater. Phys., 2009, 80, 205323.
27 W. A. Clay, T. Sasagawa, A. Iwasa, Z. Liu, J. E. Dahl, R. M. K.
Carlson, M. Kelly, N. Melosh and Z.-X. Shen, J. Appl. Phys.,
2011, 110, 093512.
28 R. Richter, D. Wolter, T. Zimmermann, L. Landt, A. Knecht,
C. Heidrich, A. Merli, O. Dopfer, P. Reiß, A. Ehresmann,
J. Petersen, J. E. Dahl, R. M. K. Carlson, C. Bostedt,
T. Mo
¨ller, R. Mitric and T. Rander, Phys. Chem. Chem. Phys.,
2014, 16, 3070–3076.
29 S. Banerjee and P. Saalfrank, Phys. Chem. Chem. Phys., 2014,
16, 144–158.
30 J. Franck and E. Dymond, Trans. Faraday Soc., 1926, 21, 536.
31 E. U. Condon, Phys. Rev., 1928, 32, 858.
32 F. Santoro, R. Improta, A. Lami and V. Barone, J. Chem.
Phys., 2007, 126, 084509.
33 V. Barone, J. Bloino, M. Biczysko and F. Santoro, J. Chem.
Theory Comput., 2009, 5, 540.
34 M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria,
M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone,
B. Mennucci, G. A. Petersson, H. Nakatsuji, M. Caricato,
X. Li, H. P. Hratchian, A. F. Izmaylov, J. Bloino, G. Zheng,
J. L. Sonnenberg, M. Hada, M. Ehara, K. Toyota, R. Fukuda,
J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao,
H. Nakai, T. Vreven, J. A. Montgomery, Jr., J. E. Peralta,
F. Ogliaro, M. Bearpark, J. J. Heyd, E. Brothers, K. N. Kudin,
V. N. Staroverov, R. Kobayashi, J. Normand, K. Raghavachari,
A. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi,
N. Rega, J. M. Millam, M. Klene, J. E. Knox, J. B. Cross,
V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R. E.
Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli,
J. W. Ochterski, R. L. Martin, K. Morokuma, V. G. Zakrzewski,
G. A. Voth, P. Salvador, J. J. Dannenberg, S. Dapprich, A. D.
Daniels, O
¨. Farkas, J. B. Foresman, J. V. Ortiz, J. Cioslowski
and D. J. Fox, Gaussian 09, Revision A.02, Gaussian Inc.,
Wallingford, CT, 2009.
35 F. Duschinsky, Acta Physicochim. URSS, 1937, 7, 551.
36 P. T. Ruhoff, Chem. Phys., 1994, 186, 355.
37 F. Santoro, FCclasses: a Fortran 77 code, 2008, available via
the Internet at http://village.pi.iccom.cnr.it, last accessed
12/02/2014.
38 F. Santoro, R. Improta, A. Lami, J. Bloino and V. Barone,
J. Chem. Phys., 2007, 126, 169903.
39 F. Santoro, A. Lami, R. Improta, J. Bloino and V. Barone,
J. Chem. Phys., 2008, 128, 224311.
40 T. Yanai, D. P. Tew and N. C. Handy, Chem. Phys. Lett., 2004,
393, 51.
41 R. Krishnan, J. S. Binkley, R. Seeger and J. A. Pople, J. Chem.
Phys., 1980, 72, 650.
42 R. W. B. Pearse and A. G. Gaydon, The identification of
molecular spectra, Chapman and Hall, 1976.
43 G. Herzberg and E. Teller, Z. Phys. Chem., Abt. B, 1933, 21,
410–446.
44 F. Marsusi, J. Sabbaghzadeh and N. D. Drummond, Phys.
Rev. B: Condens. Matter Mater. Phys., 2011, 84, 245315.
45 T. Demja
´n, M. Vo
¨ro
¨s, M. Palummo and A. Gali, J. Chem.
Phys., 2014, 141, 064308.
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