Chemical Physics Letters 442 (2007) 275–280
www.elsevier.com/locate/cplett
Precise ab-initio prediction of terahertz vibrational modes
in crystalline systems
Peter Uhd Jepsen
a,*
, Stewart J. Clark
b
a
b
COM Æ DTU, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark
Department of Physics, University of Durham, Durham DH1 3LE, United Kingdom
Received 22 January 2007; in final form 16 May 2007
Available online 3 June 2007
Abstract
We use a combination of experimental THz time-domain spectroscopy and ab-initio density functional perturbative theory to accurately predict the terahertz vibrational spectrum of molecules in the crystalline phase. Our calculations show that distinct vibrational
modes found in solid-state materials are best described as phonon modes with strong coupling to the intramolecular degrees of freedom.
Hence a computational method taking the periodicity of the crystal lattice as well as intramolecular motion into account is a prerequisite
for the correct prediction of vibrational modes in such materials.
Ó 2007 Elsevier B.V. All rights reserved.
1. Introduction
In the recent years, there has been a tremendous activity
in the field of basic and applied THz-frequency research. A
sizable fraction of this effort has been focused on the
exploitation of the fact that most organic molecules in
the solid state have a rich and distinct dielectric spectrum
in the THz region 0.3–5 THz, as exemplified by the following discussion.
It has turned out that the vibrational modes found in
this particular region of the electromagnetic spectrum are
highly characteristic not only for the molecule, but also
for its environment. For instance, the content of co-crystallized solvent molecules (e.g. water) has been shown to be
influencing factors forming the THz vibrational spectrum
of crystalline trianiline [1]. Different polymorphs of the
same molecule have also been shown to give rise to different THz absorption spectra [2], and recently the formation
of cocrystals in blends of different materials was observed
with THz spectroscopy [3].
*
Corresponding author. Fax: +45 45936581.
E-mail address: [email protected] (P.U. Jepsen).
0009-2614/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved.
doi:10.1016/j.cplett.2007.05.112
It was observed that the richly structured dielectric spectra often observed in poly- and single-crystal materials,
including powders, are due to phonon modes of the crystallites or single-crystals, whereas in amorphous materials
composed of the same molecules the spectrally localized
features are washed out and replaced by a continuous density of states [4]. Hence the long-range order of the environment of the molecules is one of the dominating influences
on the dielectric spectrum of the molecules.
The reliable prediction of the precise position and
strength of the peaks in the THz-frequency spectrum of
crystalline compounds, as well as the assignment of these
modes to specific molecular motion, has remained a major
challenge until recently for all but the simplest systems.
Exceptions are systems of high symmetry and with a small
number of atoms in the unit cell of the crystal. As examples, the phonon dispersion curves and the dielectric spectra of semiconductors with a diamond structure (Si, Ge,
and a-Sn), III–V compounds with a zincblende structure
(GaAs and related compounds), and II–VI zinc blende
compounds (ZnTe and CdTe) are well characterized [5,6].
When the number of atoms in the unit cell increases and
the interaction between the atoms in the crystal becomes
weaker than in covalent or ionic bonded crystals, the
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P.U. Jepsen, S.J. Clark / Chemical Physics Letters 442 (2007) 275–280
prediction of the phonon spectrum becomes a major theoretical and computational challenge. This is due to the
long-range, weak interactions and the large number of ions
involved in the description.
The description of solid-state, crystalline compounds
must take the periodic arrangement of atoms in the crystal
into account [7–9]. For a few, particularly simple crystalline systems, the periodic crystal structure can be approximated by linear strings of the molecules, and density
functional theory (DFT) applied to isolated clusters of
molecules can under special circumstances successfully simulate the general appearance of THz absorption spectra
[10]. However, in the general case the full crystal structure
must be taken into account in order to obtain a realistic
description of the lowest vibrational modes in a molecular
crystal. This is the fundamental reason why calculations on
isolated molecules or small units of molecules are quite successful in reproducing the mid-infrared vibrational spectra
of molecules even in the condensed phase, but fail to predict the position and intensity of low frequency modes, typically below 5–10 THz [11,12]. Therefore, calculations on
isolated molecules should be interpreted with caution when
discussing the vibrational modes in solid-state materials in
the THz region.
Important progress on simulation of solid-state THz
spectra has already been reported. Recently, Korter et al.
performed solid-state simulations, using the software packages DMOL3, CHARMM, and CPMD, predicting the THz-frequency vibrational spectra of the high explosive HMX
[13] and the amino acids serine and cysteine [14]. The same
group also published results of the solid-state calculations
of normal modes in the high explosive pentaerythritol tetranitrate (PETN) [15], using the software packages DMOL3
for the solid-state calculations. This approach has led to
convincing overall agreement between simulated and
observed line positions. A key to this good agreement is
that the periodic boundary conditions of the crystal structure is taken into account. Saito et al. compared isolatedmolecule calculations, a solid-state calculation (using the
VASP software package [16]) and experimental results of
vibrational frequencies in lactose monohydrate, and found
a good agreement between the vibrational frequencies predicted by VASP and experiment [17]. The VASP software was
also used by Saito et al. to investigate the THz vibrational
properties of the DAST crystal [18].
Here, we demonstrate that a generally applicable ab-initio simulation method, that of the plane-wave pseudopotential approach within the density functional formalism,
is capable of predicting the position and intensity, as well
as identifying the normal modes of vibrational spectra in
the THz region. The simulations are carried out without
any free empirical parameters. We demonstrate the applicability of the method with simulations of three different
hydrogen-bonded molecular crystals, the pharmaceutical
product benzoic acid, the DNA base thymine, and the saccharide sucrose. The experimentally determined THz
absorption spectra of these crystalline systems, recorded
at a temperature of 10 K, have previously been reported
[4,19,20].
2. Theory
The main goal of the calculations is to predict the low
frequency vibrational modes of molecular crystals. To
obtain accurate frequencies and spectroscopic intensities
of low frequency modes is computationally demanding
and very high convergence criteria are required throughout
the calculations [21]. When calculating the normal modes
of a molecular system, numerical errors due to convergence
tolerances are most noticeable in the low frequency modes.
Whereas an error of a few wavenumbers at high frequencies is acceptable, the same absolute error will be fatal at
the lowest frequencies.
In other work in which the low frequency THz modes of
molecular materials have been calculated, there has been a
limitation in that non-periodic molecular clusters with up
to only eight molecules have been used [10]. Although these
calculations are adequate for describing the low frequency
vibrational modes in special situations, the long range,
periodic interactions of the crystal are ignored, and additional modes due to the limited cluster size are introduced
by the calculations. To accurately predict the low frequency modes, which are essentially intermolecular in nature, the full periodic structure must be considered. In
addition to this, the long-range dipolar interaction induced
by zone center phonon modes needs to be considered
which does not form part of the cluster calculations. For
this we must turn to perturbative approaches which
includes the long wavelength limit electric field induced
by some phonon modes. Again, this is essential not only
for obtaining accurate frequencies but also their spectroscopic intensities.
The calculations are based on the plane-wave density
functional method within the generalized gradient approximation as implemented in the CASTEP code [22,23]. Normconserving pseudopotentials in the Kleinman–Bylander
[24] form are used to describe the electron–ion interactions.
The valence electron wave functions are expanded in a
plane wave basis set to a kinetic energy cutoff of 1200 eV,
which converges total energies to better than 0.1 meV/
atom. Brillouin zone integrations are performed using a
k-point set that converges the energies to an equivalent
accuracy. Electronic minimizations are performed using a
preconditioned conjugate gradient scheme [25] and are
converged to machine accuracy (approximately 1013 eV/
atom). Geometry optimizations are also performed using
a conjugate gradients scheme (the forces on the atoms were
reduced to lower than 105 eV/Å); accurate geometries
were found to be essential in order to obtain reliable values
for the low frequencies of the molecular crystals. Here the
term ‘accurate geometries’ refers to high convergence criteria as opposed to close to the experimental positions; the
forces on the atoms are very low (60.1 meV/atom) and
within 1013 eV of the potential well minimum. We find
P.U. Jepsen, S.J. Clark / Chemical Physics Letters 442 (2007) 275–280
that the ab-initio geometries are in excellent agreement with
experiment. We used X-ray diffraction results from sucrose
[26], thymine [27], and benzoic acid [28] as an input for the
atomic positions. We note that these atomic positions were
determined at room temperature, whereas our geometry
optimizations and vibrational analysis are carried out at
zero Kelvin.
Both lattice parameters and atomic positions are in
agreement to within 1% in the results reported here.
In molecular crystals there is a wide range of bonding
strengths, and therefore, to obtain the low frequency
modes accurately, all the self-consistent and the perturbative calculations along with the geometries of the system
considered must be converged to much tighter tolerances
than is usual in ‘standard’ plane-wave pseudopotential calculations. In general, we have found that the total energy
of the system (in terms of k-point sampling, total energy
convergence, etc.) must be converged to better than
106 eV/atom to finally obtain accurate low frequency
modes.
Once accurate geometric and electronic structures are
obtained, we perform density functional perturbation calculations based on the formalism of Gonze et al. [7,8] using
the CASTEP code [22,23,9]. The zone center phonon modes
are calculated and also the materials’ dielectric properties,
bulk polarizability and Born effective charges. From this
we are able to compute the spectroscopic intensities of
the modes and compare directly with experiment.
277
temperature of 10 K and with a frequency resolution of
15 GHz, obtained by using a time window of approximately 68 ps width, starting 10 ps before the main THz
signal.
4. Results and discussion
We now present the results of the THz experimental frequencies and intensities and compare with the theoretical
values. The absorption spectra of sucrose, benzoic acid,
and thymine are shown as full lines in Fig. 1, scaled with
the molar concentrations of the samples. The spectra are
recorded at a temperature of 10 K for direct comparison
with the zero-temperature ab-initio results. The frequencies
and intensities of the corresponding calculated modes are
shown as bars in the same graphs. The overall intensities
have been scaled to fit within this representation, but the
relative line strengths have not been adjusted.
The intensities of the measured absorption lines are reliable, since the spectrometer was operated below its saturation [29]. Hence not only the position but also the relative
strengths of the absorption lines can be compared with the
simulated values. This allows a very stringent test of the
simulation results.
We find a good agreement between theory and experiment in all three cases presented here, especially in the case
of sucrose. The vibrational frequencies of the sucrose crystal were calculated using the tightest convergence criteria of
3. Experimental details
Standard transmission THz time-domain spectroscopy
has been used for the experimental determination of the
dielectric function of the crystalline systems studied in this
work [19,20,4,10]. This method uses femtosecond excitation- and gate pulses in two synchronized pulse trains from
the same femtosecond oscillator to generate and to detect
ultrashort bursts of far-infrared radiation (the THz pulses).
Each of the excitation pulses drives an ultrafast current in a
photoconductive switch. The rapid acceleration dynamics
of the photogenerated charges leads to emission of a short
pulse of electromagnetic radiation. This pulse is transmitted through the sample, and detected in another photoconductive switch which is gated by a second replica of the
femtosecond pulse. By gradually changing the arrival time
of the gate pulse with respect to the THz pulse while
recording the induced photocurrent in the detector, we
can measure the temporal profile of the THz pulse with a
subpicosecond time resolution. This measurement is
repeated with and without the sample in the path of the
THz beam. A subsequent transformation to the frequency
domain and comparison of the two signals allow for the
extraction of the absorption coefficient and the index of
refraction of the sample.
The sample was placed in a closed-cycle helium cryostat
equipped with 6 mm thick polymer windows, transparent
to THz radiation. All the spectra were recorded at a sample
Fig. 1. Absorption spectrum of the molecular crystals sucrose, benzoic
acid, and thymine, in their polycrystalline forms and in the 0.7–4 THz
range.
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P.U. Jepsen, S.J. Clark / Chemical Physics Letters 442 (2007) 275–280
the three crystals studied here. The positions as well as the
intensities of each observed absorption band in the frequency region below 4 THz are in most cases reproduced
well by the DFPT simulation. The simulation predicts only
position and intensity of the normal modes of the crystal.
Hence the observed line widths cannot be compared with
our simulations.
The simulation results indicate that for instance the
broad absorption band in sucrose between 3.2 and
3.8 THz seems to be composed of several vibrational
modes. There are a few modes in the experimentally determined spectrum of sucrose that are not reproduced by the
simulation – specifically the weak features at 2.3, 2.55, and
2.6 THz. This may indicate that these observed modes are
associated with combination bands which are not
accounted for in the simulation. We note that the deficiency of the standard exchange–correlation functionals
within DFT in describing van der Waals interactions does
play a large role in these calculations.
The close agreement between experiment and theory
allows us to assign specific normal modes of the crystal
to the observed vibrational frequencies. As an example of
such an assignment, Fig. 2 shows a graphical rendering
of the predicted normal modes of the sucrose crystal at
1.601, 1.904, and 2.410 THz. The direction and relative
amplitude of the motion of each atom is indicated with
arrows.
The dimensions of the unit cell are indicated by the yellow scaffold structure, and the optimized atom placement
within the unit cell is shown. Carbon, oxygen, and hydrogen atoms are colored green, red, and gray, respectively.
The intermolecular hydrogen bonds are illustrated with
blue connections, and the intramolecular covalent bonds
are shown as gray connections. The figure confirms that
sucrose is held together by a strong intermolecular network
of hydrogen bonds. This is also the case for the other crystals studied here.
The normal-mode motions indicated in Fig. 1 allow us
to draw an important and, in our opinion, quite general
conclusion about the nature of the low frequency modes
of hydrogen-bonded molecular crystals. A pure intermolecular mode would lead to motion of the atoms of each molecule to have the same amplitude and identical or at least
highly aligned direction. However, inspection of Fig. 2
shows that this is not the case. Both direction and amplitude of the motion of each atom in the molecules is only
lightly correlated to that of the other atoms. Hence the
predicted normal mode motion is neither a pure intermolecular phonon-like motion nor a pure intramolecular
vibrational mode. In contrast there is a strong coupling
between the intra- and intermolecular motion, involving
both hydrogen bonds and covalent bonds. This illustrates
very clearly that a sharp distinction between inter- and
intramolecular modes in the THz range is not possible in
this case. Inspection of the other low frequency modes of
the sucrose crystal as well as simulation results on other
molecular crystals suggests that such a distinction is not
Fig. 2. Motion of the atoms in the sucrose crystal associated with the
predicted normal modes at 1.601, 1.904, and 2.410 THz. Hydrogen bonds
are shown in blue color while the intramolecular covalent bonds are
shown in gray color.
possible in hydrogen-bonded molecular crystals, even for
the lowest-frequency modes. In non-hydrogen-bonded
molecular crystals, where the intermolecular interactions
are weaker than in hydrogen-bonded molecular crystals,
there may be a more distinct separation between interand intramolecular modes.
P.U. Jepsen, S.J. Clark / Chemical Physics Letters 442 (2007) 275–280
279
Fig. 3. Phonon-like contribution (curves shown in black color) and intramolecular contribution (curves shown in red color) to the first 50 normal modes
of benzoic acid, sucrose, and thymine.
In Fig. 3, we introduce a method to quantify this statement for the molecular crystals considered in this work.
For each vibrational mode, the motion of the center of
mass of each molecule in the unit cell is a measure of the
phonon character of the mode. On the other hand, the
motion of each atom in the molecules relative to the
motion of the center of mass of each molecule is a measure
of the intramolecular character of the mode. More precisely, for the first 50 normal modes of each molecular crystal we have plotted the quantities
1 X
DCM ¼ mi di ;
ð1Þ
M
X
1
ð2Þ
DRM ¼
jdi DCM j;
N
where M is the mass of the molecule, N is the number of
atoms in the molecule, and di is the displacement vector
of each atom for the particular mode. Fig. 3 shows that
for each of the crystals considered here, each vibrational
mode has considerable contribution both from phononlike motion (DCM > 0) and from intramolecular motion
(DRM > 0).
There is an additional complication in calculating the
vibrational frequencies of zone center modes, in that we
must also consider the LO/TO splitting which occurs and
is due to the finite electric field created by excitation of phonons which break the center of symmetry and create a nonzero dipole per unit cell. We have evaluated the electric
field response giving the change in mode frequency, which
is included in the diagrams. In addition to the vibrational
frequencies and their intensities, we have calculated the
bulk polarizabilities and permittivities which are given in
Tables 1 and 2 respectively in both the high and low frequency limits.
Fig. 4 shows the real part of the dielectric function of
polycrystalline sucrose, recorded at 10 K and extracted
Table 1
Optical polarizabilities aopt and bulk polarizabilities aDC
aopt
(Å3)
Benzoic acid
Sucrose
Thymine
79.99
0.00
7.35
79.23
0.00
1.13
81.97
0.00
12.84
aDC
(Å3)
0.00
67.01
0.00
0.00
78.54
0.00
0.00
80.43
0.00
7.35
0.00
73.70
1.13
0.00
83.91
12.84
0.00
46.43
96.09
0.00
10.69
150.27
0.00
16.83
78.44
0.00
1.60
0.00
109.59
0.00
0.00
145.50
0.00
0.00
132.74
0.00
10.69
0.00
97.54
16.83
0.00
139.75
1.60
0.00
68.29
0.00
3.23
0.00
0.00
3.55
0.00
0.00
3.93
0.00
0.21
0.00
2.98
0.29
0.00
3.45
0.03
0.00
2.50
Table 2
Optical permittivities 1 and static permittivities DC
1
Benzoic acid
Sucrose
Thymine
2.62
0.00
0.14
2.39
0.00
0.01
2.81
0.00
0.28
DC
0.00
2.36
0.00
0.00
2.38
0.00
0.00
2.77
0.00
0.14
0.00
2.50
0.01
0.00
2.47
0.28
0.00
2.02
2.95
0.00
0.21
3.64
0.00
0.29
2.73
0.00
0.03
from the same THz-TDS data set as used in Fig. 1a
( = n2 j2, j = ac/2x). From this plot, the static permittivity of sucrose can be estimated, and we obtain
DC 3.28. This value is lower than the average of the
diagonal components of the calculated relative permittivity
tensor for sucrose in Table 2. The reason for this discrepancy is that the sample material was obtained from pressed
powder, and therefore, the sample had lower density, and
hence, lower index of refraction, than single-crystalline
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P.U. Jepsen, S.J. Clark / Chemical Physics Letters 442 (2007) 275–280
anharmonicity in other biologically relevant molecules,
such as biotin [30].
Acknowledgement
We acknowledge partial financial support from the EU
project TeraNova and from the Danish Research Agency.
References
Fig. 4. The real part of the dielectric function of polycrystalline sucrose,
recorded at a temperature of 10 K.
material. The dielectric function at the high end of the THz
spectrum in Fig. 4 is higher than the predicted value of 1.
This is expected since the molecular modes at higher frequencies will lead to a lowering of the permittivity in the
high-frequency limit.
5. Conclusions
We have demonstrated that the plane-wave DFPT
method taking periodic boundary conditions into account,
is capable of simulating the THz vibrational spectra of different hydrogen-bonded crystals with convincing accuracy.
The simulations are carried out with the CASTEP code
implementation of DFPT, and are performed without
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in hydrogen-bonded crystals there is a strong coupling
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The detailed information available with the simulation
method presented here about the solid-state THz vibrational modes of molecules will enable researchers to obtain
important insight into the weak, delocalized forces that
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taking the temperature of the crystal into account, and
hence, investigate the temperature-dependent interplay
between weak and strong intermolecular forces. Similar
studies could also further aid the understanding of THz
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