Post on 08-Apr-2015
description
修士論文
Ab Initio Dynamics Simulation
of the Molecular Gyroscope
(分子ジャイロスコープの第一原理動力学
シミュレーション)
Anant Babu Marahatta
平成21年
Graduate school of Science
Department of Chemistry
Tohoku University, Japan
1
Contents
Acknowledgement
Abstract
CHAPTER 1
1 INTRODUCTION 1-21
1.1 Macroscopic Gyroscope 1
1.2 Molecular Gyroscope 2
1.3 Experimental information 6-12
1.3.1 Synthetic Details 6
1.3.2 Analytical Details 6
A X-ray analysis 6
B 13C CP/MAS NMR Spectroscopy 8-
1 Basic principle and methodology 8
2 Spectra analysis and confirmation of phenylene rotation
10
2 OBJECTIVE OF THIS STUDY 13-14
3 THEORETICAL BACKGROUND 15-21
3.1 Density Functional based Tight Binding [DFTB] method 15
3.2 Molecular Dynamics (MD) Simulation 18
3.3 Velocity Verlet Dynamics 20
CHAPTER 2
RESULTS AND DISCUSSION 22-42
2.1 Dynamics of an Isolated Molecular Gyroscope 22
2.1.1 Gaussian approaches: Full optimization case 22
A Rotational Potential Energy Surface 23
2.1.2 Gaussian approaches: Single Point (SP) energy calculation case 25
A Computational procedure 25
B Rotational Potential Energy Surface 27
2.1.3 DFTB approaches: Full optimization case 28
2
2.1.4 DFTB approaches: Static calculation case 29
A Computational Procedure 30
B Rotational Potential Energy Surface 30
2.2 Dynamics of the Molecular Gyroscope under crystal conditions 31
2.2.1 Gaussian approaches 32
2.2.2 DFTB approaches: Static calculation 33
A Computational procedure 33
B Rotational Potential Energy Surface 33
2.2.3 DFTB approaches: Full optimization case
A Computational procedure 37
B Rotational Potential Energy Surface 37
2.3 DFTB Molecular Dynamics (MD) Simulation 39
2.3.1 Rotary motion of the phenylene group 40
A Low temperature case 40
B High temperature case 41
2.4 Summary 42
BIBLIOGRAPHY 45
APPENDIX-I 48-60
DFTB code of an isolated molecule for the static calculation 48
3
ACKNOWLEDGEMENT
I would like to express my sincere gratitude to my supervisor, Prof. Hirohiko Kono,
Department of Chemistry, Tohoku University, for his kind help, valuable guidance and
encouragement throughout this research work without which this work would not be possible.
I am greatly indebted to Prof. Wataru Setaka, Department of Chemistry, Tokushima
Bunri University, for his fruitful discussion and also providing me the the X-ray geometry and 13C CP/NMR MAS spectra of the concerned Molecular Gyroscope.
I would also like to express my deep appreciation to Prof. Yuichi Fujimura and A. prof.
Yukiyoshi Ohtsuki, Department of chemistry, Tohoku University, for their encouragement
during the work.
I would like to acknowledge and extend my heartfelt gratitude to Dr. Kunihito Hoki,
Department of Chemistry, Tohoku University, for his continuous support and valuable guidance
throughout this research work. Without his encouragement and constant guidance, I could not
have finished this dissertation.
I would also like to extend my sincere thanks to Ms. Chieko Azuma, lab secretary, for her
proper management and caring about my academic parts during this work.
I am also highly grateful to my seniors Dr. Manabu Kanno, Walid M. I. Hassan,
Toshihiro Yamada and Naoyuki Niitsu for their kind help while handling computer software. At
last but not the least, I am very much thankful to all the colleagues of “Mathematical Chemistry
Laboratory” for their kind cooperation during this dissertation.
Anant Babu Marahatta
Department of chemistry
Tohoku University, Japan
2009 August
4
Abstract
The molecular dynamics calculation of the molecular gyroscope which has a
phenylene rotor encased in three long siloxaalkane spokes is performed for the first
time by using the ab initio techniques like Hartree-Fock [HF] method, Density
Functional Theory [DFT] and the semi-empirical approach such as Density
Functional based Tight Binding [DFTB] theory.
The validity of the DFTB method is checked in reference to this molecular
gyroscope. It is confirmed that DFTB can reproduce the main features of the
potential energy surface obtained by the conventional DFT such as B3LYP.The
optimized structure obtained from DFTB method agreed well with the X-ray
observation except the flexible Si-O-Si angles in the siloxaalkane spokes.
Furthermore, transition states of the rotational motion of the phenylene group
under periodic boundary condition were also obtained, where the highest activation
energy of the rotation was found to be around 500 cm−1 which is almost four times
greater than that of the isolated molecule obtained from the B3LYP/6-31G level of
calculation. Thus the phenylene rotor is found to interact strongly with the periodic
molecular array during the rotation.
The results of the MD simulation under DFTB show that the stable
structures of the molecule are appeared at the same angle of phenylene rotation to
that observed at the potential energy surface derived by the DFTB with periodic
boundary condition. It also indicates that at room temperature, the phenylene rotor
stays around the stable position at least for 1ns. However, at high temperature of
about 1200 K, phenylene rotor undergoes flipping in an average time of 20ps. This
flipping motion at high temperature indicates the facile phenylene rotation of the
siloxaalkne molecule in solid state as observed by the X-ray diffraction technique.
5
CHAPTER 1
1. Introduction
1.1 Macroscopic Gyroscope
Invented since 1917, the original Toy Gyroscope has been a classic
educational toy for the discovery of modern macroscopic gyroscopic. A common
macroscopic gyroscope is a device consisting of a spinning mass, or rotor, with a
spinning axis that projects through the center of the mass, which is mounted within
a rigid frame, or stator. It is used as a navigational device to measure or maintain
orientation in ships, aircrafts, spacecrafts, vehicles etc. It works on the principle of
the conservation of angular momentum [1]. The fundamental parts involve during
rotation are labeled in fig.1 shown below.
The rotor, whose
center of mass is in a fixed
position, spins
simultaneously about one
axis and is capable of
oscillating about the two
other axes, and thus it is
free to turn in any direction
about the fixed point. The
behavior of a gyroscope can
be most easily appreciated
by the consideration of the front wheel of a bicycle. If the wheel is leaned away
from the vertical so that the top of the wheel moves to the left, the forward rim of
6
the wheel also turns to the left. In other words, rotation on one axis of the turning
wheel produces rotation of the third axis. In analogy with this macroscopic
gyroscope and to suggest some of their properties and functions, the molecular
structure is referred to molecular gyroscopes [2].
1.2 Molecular Gyroscope
Molecular scale machinery provides an avenue for the study of nano-science
and for exploring its applications. Since the mechanical machines are able to create
motion, produce work, pump heat or perform other useful functions, the molecular
machine is also required to integrate molecular assemblies so as to achieve such
performance. An example of such molecular machine is the molecular gyroscope
[1].
Complex dynamics in high-density machines such as automobile engines,
typewriters, mechanical clocks, etc. rely on volume-conserving periodic processes.
These volume-conserving molecular motions have already been well documented
in crystalline solids [3]. The first step to realize a macroscopic object at the
molecular level is to select the atomic and molecular components which approach
the desired structure and its function. To generate motion, a machine has to consist
of moving parts and requires at least one source of energy.
The central rotating part of a molecular gyroscope may be any symmetric
group with its center of mass aligned along a single bond that supplies both the
rotary axis and the point of attachment to the static framework. The stator should
provide an encapsulating frame to shield the rotor from steric contacts with
adjacent molecules in the crystal. Dipolar molecular rotors sometimes referred to
molecular compasses; have a subgroup containing a permanent electric dipole
moment that rotates relative to another part of the molecule. Alternatively, the
function of crystalline molecular machines may rely on the collective response of
7
reorienting dipoles to the presence of electric, magnetic and photonic stimuli [3].
By introducing a dipole moment, which interacts with electric fields, on the rotor,
the unidirectional motion of it could be controlled and functionalized by a static
electric field [4]. Molecular gyroscopes and compasses provide one of the most
promising structural designs.
Although the realization of molecular analogs of macroscopic gyroscopes
presents serious challenges and limitations, the molecular array having close
association may lead to several interesting applications. Molecular gearing systems
are the first successful examples of rotary molecular devices engineered and
synthesized using conventional chemistry [5]. As the methods to measure the bulk
macroscopic viscosity are well developed, imaging local microscopic viscosity
remains a challenge, and viscosity maps of microscopic objects, such as single
cells, are actively sought after. A new approach to image local micro-viscosity
using the fluorescence lifetime of a molecular gyroscope is recently reported [6].
The variety of fluorescent molecular rotors has been developing to report on
specific cell targets.
Similarly molecular rotors driven by LASER pulses are also widely reported
[7]. It is expected that the molecular gyroscope has tremendous potential
applications in LASER industries. By adsorbing the stator of the molecular
gyroscope, by some chemical means, at the surface of the goggles, the unwanted
LASER beam can be blocked. Thus Molecular gyroscope is also accepted as one
of the safety devices. In addition, arrays of rotors could propagate molecular rotary
waves at speeds much lower than typical phonon velocities. This behavior might
have application to radio frequency filters [1].
Some of the challenges and limitations at the molecular level include the
construction of frictionless rotors, the need for flat (or barrierless) potential energy
surfaces, mechanisms to introduce a controlled impulse or a constant force to
8
power the rotor, and, most importantly, the fact that momentum and energy are
internally redistributed within a few picoseconds in dissipative molecular systems,
making it very challenging to induce unidirectional rotation [8].
As a class of molecular machines, much attention has been focusing on
macrocyclic molecules with bridged phenylene groups, because they are expected
to demonstrate functions of molecular gyroscopes and compasses, whose interior
rotator (phenylene) is protected by an exterior framework . Garcia-Garibay et al.
has first proposed a triply bridged 1,4 bis[(tritylethynyl)-2,3-difluorobenzene
shown in fig 2(a), as a solid-state molecular gyroscope [2, 9]. Moreover, they have
also modified the previous molecular gyroscope and studied the dipolar rotor-rotor
interactions in molecular rotor crystal of a 1,4-bis(3,3,3-triphenylpropynyl)-2-
fluorobenzene molecule shown in fig 2(b) .
Similarly, the novel molecular gyroscope, shown in fig 3, having a
phenylene rotor encased in three long siloxaalkane spokes has recently been
synthesized by Setaka et al [10]. The rotary motion of the phenylene has also been
observed by an X-ray analysis and 13C CP/MAS NMR spectroscopy. During the
course of our studies, for the first time, an “ab initio molecular dynamics
simulation” of this siloxaalkane rotor is carried out in order to explain the
dynamics of rotation in more detail.
9
10
Fig. 3 An X- ray structure of Siloxaalkane rotor. Gyroscopic parts are labeled.Hydrogen atoms are omitted for clarity.
1.3. Experimental information
1.3.1 Synthetic Details
A molecular gyroscope with a phenylene rotor encased in three long
siloxaalkane spokes is synthesized by Setaka et. al by using commercially
available reagents [10]. The synthetic details are given at the supplementary
material of this journal. The authors have mentioned that the percentage yield of
the siloxaalkane rotor is only about 38% whereas that of the byproduct is about 54
%. Eventhough the percentage yield is low, the siloxaalkane rotor is found as a
potential gyroscopic molecule. It is also reported that the byproduct formed is one
of the isomers of the siloxaalkane rotor which does not behave like the first one.
1.3.2 Analytical Details
The identification of the synthesized molecular compound is carried out by an
X-ray crystallography and 1H, 29Si and 13C NMR spectroscopies. The Empirical
formula and formula weight are identified as C54H124O3Si14 and 1214.79
respectively. The analytical details are also reported at the supplementary material
of this journal [10]. The properties observed by the authors are listed below.
A. X-ray analysis
An X-ray structure of Siloxaalkane rotor with its three stable positions is shown
above in fig. 3 [10]. It is noted that the structure of the crystal strongly depends on
the temperature. The structure is changed from triclinic to monolinic when the
temperature increases from 173 K to 223 K. The observed lattice parameters are
listed on the following table 1. In triclinic crystal geometry, the molecules are
11
found to arrange with the parallel axes of rotation. The wave length of the X-ray
used during analysis is noted as 0.71070 Å in both temperatures.
At 173 K, the strong deformation of the conformation of siloxaalkane spoke is
noted in comparison to that at 223 K. Such deformation causes the reduction in the
unit cell volume followed by the significant modification of the phenylene disorder
due to an increase in the steric contact between phenylene rotor and surrounding
arms.
Table 1. An X-ray analysis of Silloxaalkane rotor
S.No. Temperature(K)
Crystal Structure andUnit cell dimensions (Å)
Unit cell Volume (Å3)
Space group
1 173 Triclinic 4051 P-1a=11.818 α=91.852b=14.552 β= 99.156c=23.876 γ= 90.174
2 223 Monoclinic 4133 Pna= 11.840 α=90.0b= 14.619 β= 99.240c= 24.188 γ= 90.0
The site occupancy factors of the phenylene rotor at three observed stable
positions are also reported at these two temperatures. The strong temperature
dependent site occupancy factors are observed. Thus it is stressed that the rotary
motion of the phenylene is also temperature dependent.
The reduction of the area and the strong temperature dependent site
occupancy factors suggested that the phenylene ring rotates smoothly at 223K and
above but flips in a confined area at 173 K and below.
12
B. 13C CP/MAS NMR Spectroscopy
1. Basic principle and methodology
The nuclei of many elemental isotopes have a characteristic spin (I). Some
nuclei have integral spins (e.g. I= 1, 2, 3 ....), some have fractional spins (e.g. I =
1/2, 3/2, 5/2 ....), and a few have no spin, I = 0 (e.g. 12C, 16O, 32S....). Isotopes of
particular interest and use to organic chemists are 1H, 13C, 19F and 31P, all of which
have I = 1/2. Since the analysis of this spin state is fairly straightforward.
For nuclei of spin 1/2, signals broad enormously and makes complication to
interpret the spectra. This line broadening is due to the chemical shift anisotropy of
frequency about 103 to 104 Hz and anisotropic dipolar coupling of frequency about
20×103Hz.
The chemical shift anisotropy is governed by an inductive effect. If the
electron density about a 13C nucleus is relatively high, the induced field due to
electron motions will be stronger than if the electron density is relatively low. The
shielding effect in such high electron density cases will therefore be larger, and a
higher external field (Bo) will be needed to excite the nuclear spin. Such nuclei are
said to be shielded. The exactly opposite phenomenon is known as deshielded. The
deshielding effect of electron withdrawing groups is roughly proportional to their
electronegativity.
Just like the chemical shift interaction, the dipolar coupling is also one of the
potential complications in NMR spectroscopy. This coupling summarizes the
energy relationship between two NMR active nuclear spins like 13C and 1H. However, couplings between neighboring carbons can be ignored due to the
low natural abundance of ~1.1% of 13C [11, 12]. The dipolar coupling depends on
13
the orientation of the internuclear-spin vector with respect to the axis of the applied
magnetic field [13] and creates a “J-coupling”. A “J-coupling”, some time called an
indirect “dipole dipole coupling”, is the coupling between two nuclear spins due to
the influence of bonding electrons running between the two nuclei on the magnetic
field.
The dipolar coupling can be decayed by the dipolar dephasing experiments.
The dipolar dephasing utilizes differences in the strength of 13C-H dipolar coupling
to enable the distinction between protonated and non-protonated carbons, and
molecularly mobile and rigid carbons. During the experiment, the high power
proton decoupler is turned off for a short period (the dephasing delay) between
polarization and detection, during which time 13C signal is lost. However molecular
motion is a complicating factor. Despite the presence of three attached protons,
methyl groups undergo dephasing slowly, because rapid rotation greatly
diminishes the strength of the coupling. Decreased rates of dephasing also occur
for other moieties with high degrees molecular motion, such as lipids near their
melting points [14].
In order to suppress these couplings, which would otherwise complicate the
spectra and further reduce sensitivity, 13C-NMR spectra are proton decoupled by
using Cross Polarization Magic Angle Spinning [CP/MAS] experiment. Solution-
like spectra can be measured with solid samples by taking advantage CP/MAS
developed by Pines and co-workers [15, 16]. It is known that 13C nucleus is over
fifty times less sensitive than a proton in the NMR experiment. Thus cross
polarization method is required to transfer the magnetization from the highly
abundant and sensitive hydrogen atoms to the less sensitive and highly diluted 13C
nuclei. In brief, CP/MAS experiment includes simultaneous high power 1H-
decoupling and fast sample spinning (5-20 kHz) at the magic angle (54.7˚) to
14
remove the line broadening that comes from static anisotropic interactions
mediated by the external magnetic field.
It is well known that the signal originating from the species which have
strong coupling to hydrogen decay [dephase] faster than the signal originating from
species with weak coupling to hydrogen. Thus by interrupting decoupling between 13C and H after cross polarization, the signals corresponding to strongly coupled
protonated carbons become disappear. As the dephasing relies on the dipolar
coupling, anything that reduces it can result in signals having slow decay rate. The
signal intensity for strongly coupled carbons usually decays within 50 μs and the
molecular structure greatly influences the effective dipolar interactions [17]. The
classic example is methyl groups where internal rotation reduces the dipolar
coupling between 13C and attached H which results in methyl signals rarely
suppressed. Changing the dephasing rate of the signal of 13C during flipping of the
Phenyl ring is another well known example [17, 18, 19 ].
2. Spectra analysis and confirmation of phenylene rotation
In order to probe the rotation of phenylene rotor in siloxaalkane gyroscope,
Setaka et. al have applied the 13C CPMAS NMR spectroscopy tool to
siloxaalkane rotor as a target molecule and 1,4-bis (tri-methylsilyl) benzene as a
reference molecule [10]. The structure of the reference molecule and the molecular
model of the siloxaalkane rotor are shown in fig. 4(a) and 4(b) respectively.
The model of the siloxaalkane rotor is designed in order to compare 13C
CP/MAS NMR spectra of the central phenylene part between it and reference
molecule. The phenylene ring of both of them is bonded to the two silicon atoms
via 1 and 4 axial carbon atoms. However, each silicon atom of the reference
molecule is bonded with three methyl groups but that of the siloxaalkane rotor is
15
bonded with three long siloxaalkane arms represented by the arcs in concerned
figure. Each arc represents a chain of -C-C-Si-C-C-Si-O-Si-C-C-Si-C-C- atoms
with two atomic Hydrogen bonded at each Carbon and two CH3 groups at each
Silicon atom.
The observed 13C CP/MAS NMR spectra at 298 K are shown in fig.5. The
first signal of both of these spectra at around 135 ppm is due to two 13C atoms
bonded with two silicon atoms whereas the second signal around 131 ppm is due to
the 13C atoms bonded with one hydrogen atom each. As the dephasing time
increases, the signal at around 131 ppm in fig. 5(b) is rapidly disappeared with in
60 μs but the corresponding signal remains intense in fig. 5(a) even upto 120 μs
dipolar dephasing delay. However, the signals arise due to two axial 13C atoms
bonded with silicon atom remain unaffected in both of these cases.
As the aromatic 13C-H signal of the reference molecule is disappeared
around the “threshold dephasing delay time” (~50 µs), one can conclude that no
disturbance is created on dipolar coupling between 13C and bonded H. Thus the
16
Fig. 4(b) Model of the Siloxaalkane rotor. Each arc represents the stator.
Fig. 4(a) Molecular structure of the reference molecule. Methyl groups bonded to Si are not shown.
usual decay rate of the aromatic 13C-H signal indicates no rotation of the phenylene
in the reference molecule. Whereas appearance of the intense signal even after
crossing the “threshold dephasing delay time” confirms the slow decay rate of the
aromatic 13C-H signal. The slow decay rate is governed by the weakening of
dipolar coupling between 13C and H nuclei which is further due to the rotation of
the phenylene rotor in the solid state.
17
Fig. 5(a) Spectra of the siloxaalkene gyroscope with dipolar dephasing delay at 298 K.
Fig. 5(b) Spectra of a reference molecule with dipolar dephasing delay at 298K
.
2. Objective of this study
Recent synthesis of the molecule with phenylene ring encased in three long
siloxaalkane arms and the characterization of it by 13C CP/MAS NMR and X-ray
diffraction as a gyroscopic in nature influences us to carry out further research.
The sole objectives of this study are:
To confirm the X-ray structure of the compound by quantum chemistry
calculations.
To understand the dynamics of the rotation confirmed by 13C CP/MAS NMR
spectroscopy and X-ray analysis.
As this molecule exists in monoclinic crystalline geometry with two
molecules per unit cell at 223 K and above, it is mandatory to perform the
computational calculation under the crystal condition. Exploring the interactions
between the rotating molecule and its surroundings, which determine the molecular
dynamics, is the key point of this work. But for the computational chemists,
consideration of the periodic boundary condition with two molecules of each
containing 195 atoms per unit cell is really a big system.
Since, there are accurate ab initio calculations based on density-functional or
Hartree-Fock theory, which represent without any doubt a very reliable benchmark
for all other methods. In contrary, these methods are too slow and some time
inefficient for the investigation of many interesting properties of the larger
crystalline systems. So, a cheap yet decent chemical model becomes our choice for
studying dynamics.
18
To achieve this goal with reasonable computational costs, Density
Functional based Tight Binding (DFTB) method [20] is applied to the siloxaalkane
gyroscope and evaluated its validity and strength in reference to this molecule.
19
3. Theoretical background
3.1 Density Functional based Tight Binding [DFTB] method
Besides the traditional quantum chemical ab initio methods based on the
Hartree-Fock scheme plus a proper treatment of the electron correlation, now the
density functional theory (DFT) in the realization of Kohn and Sham (KS) [21]
have become well established in studies of the electronic structure and structure of
molecules, clusters, and solids.
Such calculations allow the study of rather large systems with a reasonable
computational effort and a quite good accuracy of the results. Furthermore, it also
allows the study of dynamical processes [20 (b)]. However, there are still many
systems that are too large to be studied by full ab initio techniques, such as large
clusters, biomolecules, or solids with very large unit cells or even amorphous
solids.
Moreover, to study the dynamics of complex systems over a long simulation
time requires approximate schemes. Such approximate method so called DFTB is
one of the widely used treatments. It has a semi empirical approach to some extent
as well. The use of only a few semi empirical parameters minimizes the effort for
the determination of them; it yields a close relation to full ab initio DFT schemes
and it guarantees a good “transferability” of the parameters, going from one system
to another. On the other hand, the use of some approximations in connection with a
few empirical parameters makes the scheme computationally extremely fast [20,
21].
Mathematically, DFTB method is an approximate Kohn-Sham density
functional theory (KS-DFT) scheme with an LCAO representation of the KS
orbitals. After the development of a non-self-consistent (“zeroth order”) approach,
20
a self consistent charge (SCC) extension was formulated with an extension to the
consideration of spin polarization [23].
The LCAO treatment allows to write nth MO, Ψn(r) represented by using
valence AO, Φia(r) as
(1)
where,
i :Index of an atom
a :Index of an orbital
C : superposition of the valence orbitals
Tight binding (TB) secular equation is defined as
(2)
Hia,jb :TB Hamiltonian Matrix
Sia,jb :Overlap matrix
ε :Eigen value
The matrix elements of the Hamiltonian are defined by using the effective
Kohn-Sham potential Veff(r) as
(3)
where,
21
(4)
The effective Kohn-Sham potential is approximated as a simple superposition of
the potentials of the neutral atoms Vi0:
(5)
By applying the two-center approximation, the Hamiltonian matrix element
becomes
iα= jβ
i ≠ j (6)
Otherwise
The approximations formulated above lead to the same structure of the
secular equations as in tight-binding (TB) but it has an important advantage that all
matrix elements are calculated within the density functional theory and none of
them is handled as an empirical parameter [20 (b)]. In principle, there are no
empirical parameters, instead, all quantities are either calculated within DFT or
they are determined in reference structures by DFTB calculations [23].
Because of keeping the essential features of DFT, the DFTB method is
called as an approximate DFT scheme. Without having a large number of empirical
parameters, it has the efficiency of traditional semi-empirical quantum chemical
methods.
22
3.2 Molecular Dynamics (MD) Simulation
Molecular dynamics is a form of computer simulation in which atoms and
molecules are allowed to interact for a period of time by approximations of known
physics, giving a view of the motion of the atoms. It was originally conceived
within theoretical physics in the late 1950s [24] and early 1960s [25], but is applied
today mostly in materials science and modeling of biomolecules.
Because molecular systems generally consist of a vast number of particles, it
is impossible to find the properties of such complex systems analytically. MD
simulation solves this problem by using several numerical algorithms. It represents
an interface between laboratory experiments and theory, and can be understood as
a "virtual experiment". It also probes the relationship between molecular structure,
movement and function under the periodic boundary condition. The obvious
advantage of MD simulation over another widely used Monte Carlo [MC]
simulation is that it gives a route to dynamical properties of the system like time-
dependent responses to perturbations, rheological properties and spectra etc.
MD simulation generates information at the microscopic level, including
atomic positions and velocities. The conversion of this microscopic information to
macroscopic observables such as pressure, energy, heat capacities, etc., requires
statistical mechanics. The connection between microscopic simulations and
macroscopic properties is made via statistical mechanics which provides the
rigorous mathematical expressions that relate macroscopic properties to the
distribution and motion of the atoms and molecules of the N-body system. MD
simulations provide the means to solve the equation of motion of the particles and
evaluate these mathematical formulas.
MD has also been termed "Laplace's vision of Newtonian mechanics"
because of predicting the future by animating nature's forces and allowing insight
23
into molecular motion on an atomic scale [26, 27]. This computational method
calculates the time dependent behavior of a molecular system. MD simulations are
also used in the determination of structures from X-ray crystallography and from
NMR experiments [28]. To reproduce the dynamics of molecular systems, proper
selection of algorithms and parameters are very important. Furthermore, current
potential functions are not sufficiently accurate in many cases, so the much more
computationally demanding ab Initio Molecular Dynamics method has been using
widely.
In MD simulation, the interaction between the particles is described by a
"force field". The force created by electrons make the movement of the nuclei
based on the classical mechanics. The MD simulation method is based on
Newton’s second law or the equation of motion, F=ma, where “F” is the force
exerted on the particle, “m” is its mass and “a” is its acceleration. From the
knowledge of the force on each atom, it is possible to determine the acceleration of
each atom in the system. Integration of the equations of motion then yields a
trajectory that describes the positions, velocities and accelerations of the particles
as they vary with time. From this trajectory, the average values of properties can be
determined. The method is deterministic; once the positions and velocities of each
atom are known, the state of the system can be predicted at any time in the future
or the past.
Since Newton’s second law preserves the total energy of the system, and a
straightforward integration of Newton’s second law therefore leads to simulations
preserving the total energy of the system (E), the number of molecules (N) and the
volume of the system (V). That’s why it is widely called as an NVE simulation.
Since the potential energy is a function of the atomic positions (3N) of all the
atoms in the system. Due to the complicated nature of this function, there is no
24
analytical solution to the equations of motion; they must be solved numerically
[19]. The most appropriate algorithm for doing this is a velocity verlet dynamics
[29].
3.3 Velocity Verlet Dynamics
Solving Newton's equations of motion does not immediately suggest
activity at the cutting edge of research. The molecular dynamics algorithm in most
common use today may even have been known to Newton [30]. The most
commonly used time integration MD algorithm is probably the so-called Verlet
algorithm [29]. The basic idea of it is to write two third-order Taylor expansions of
the position vectors r (t) in different time.
Since we are integrating Newton's equations, a(t) is just the force divided by
the mass, and the force is in turn can be expressed as the gradient of the potential
energy which in turn a function of the positions r(t):
(7)
An even better implementation of the same basic algorithm is the so-called
velocity Verlet scheme, where positions, velocities and accelerations at time t+∆t
are obtained from the same quantities at time t in the following way:
(8)
(9)
(10)
(11)
25
Choosing the time step dt is essential to success. Too large, and errors will
accrue in the integration. Too small, and errors will occur from rounding in the
computation.
26
Chapter 2
Results and Discussion
2.1 Dynamics of an Isolated Molecular Gyroscope
2.1.1 Gaussian approaches: Full optimization case
It is general expectation to start the computational calculation from
Hartree-Fock theory with minimal level of basis set. Thus in this case too, before
carrying out other related calculations, the X-ray geometry is fully optimized by
HF/STO-3G level of calculation.
It is observed that the optimized structure is far from the X-ray structure in
several approaches. The major changes happen on the Si-O-Si bond angle and the
orientation of the arms as shown in table 2 and 3 respectively. In comparison to the
X-ray structure, the distance between Oxygen atoms of each arm is increased,
whereas the Si-O-Si bond angle in each arm is reduced. The optimized structure
has siloxaalkane arms with more acute Si-O-Si bond angle in comparison to that in
X-ray structure. However, the accuracy on the structure is increased by increasing
the richness of the basis set. As mentioned in table 2 and 3, the Si-O-Si bond angle
and the position of the arms are far better in HF/6-31G level of calculation in
comparison to the HF/STO-3G calculation.
Moreover, the X-ray geometry is also optimized by using B3LYP
calculation with 6-31G basis set. The optimized structure seems very close, in
several aspects, to that of the X-ray structure in comparison to the Hartree Fock
calculation.
27
Table 2: Variations in the Si-O-Si bond angles
Structure (isolated) Si-O-Si bond angle in each arm (degree)X-ray 169.4 172.6 162.9Optimized[HF/STO-3G] 137.5 136.8 137.4Optimized [HF/6-31G] 176.5 171.3 171.3Optimized [B3LYP/6-31G] 174.7 164.7 164.2
Table 3: Variations in the distance between Oxygen atoms of each arm
Structure (isolated) Distance between Oxygen atoms of each arm (A˚)X-ray 8.99 8.67 8.50Optimized[HF/STO-3G] 10.55 9.75 9.75Optimized [HF/6-31G] 10.25 9.96 9.96Optimized [B3LYP/6-31G] 10.14 9.66 9.54
For more comparative purposes, the X-ray structure of the isolated molecule
is also optimized by DFTB package as well.
A. Rotational Potential Energy Surface
The way the energy of a molecular system varies with small changes in its
structure is specified by its potential energy surface. A potential energy surface is a
mathematical relationship linking molecular structure and the resultant energy
[31].Behind the analogies of above section 2.1.1, the nature of the rotational
potential energy surface is calculated by using these Gaussian approaches with the
advantage of Opt=Modredundant option [32]. The central phenylene rotor is
rotated by changing a dihedral angle and the potential energy of the fully optimized
structure at each angle is calculated. This energy is plotted against the angle of
rotation as shown in fig. 6.
28
100
80
60
40
20
0
-20
Pot
entia
l Ene
rgy
(cm
-1)
350300250200150100500Angle of rotation (deg.)
HF/STO-3G calculation
The appearance of the six minima in fig. 6(a) shows the six folded symmetry
of the phenylene rotor. Fig. 6(b) shows that the rotational energy barrier is found to
29
Fig. 6(a) Rotational potential energy surface
Fig. 6(b) Rotational potential energy surface upto 60˚ rotation
140
120
100
80
60
40
20
0
Pot
entia
l Ene
rgy
(cm
-1)
6050403020100Angle of rotation (deg.)
Red line: HF/ STO-3GBlue line: HF/ 3-21GGreen line: HF/ 6-31GPink line: B3LYP/ 6-31G
be around 140 cm-1 for the HF/STO-3G calculation whereas the surface becomes
more flat after increasing the richness of the basis set. The barrier becomes around
100 cm-1 when the calculation level is B3LYP/6-31G. Thus the isolated molecule
seems as almost free rotor. However, it is needful to consider the periodic arrays of
the molecules to explain real dynamics in solid state.
2.1.2 Gaussian approaches: Single Point (SP) energy calculation case
A single point energy calculation is the prediction of the energy and related
properties for a molecule with a specified geometric structure. In other words, the
sum of the electronic energy and nuclear repulsion energy of the molecule at the
specified nuclear configuration can be evaluated by single point energy calculation
without doing optimization [33]. The validity of results of these calculations
depends on having reasonable structures for the molecules as input [31]. Thus, in
order to maintain the X-ray geometry during the quantum chemistry calculation, it
is mandatory to perform the single point energy calculation.
In the absence of an experimentally derived potential energy surface [PES],
for the computational chemists, the surface derived from the SP calculation acts as
a right hand to probe the dynamics. Thus to compare the result obtained from the
full optimization case of Gaussian as explained above, sketching PES with respect
to SP energy calculation seems very necessary. The general computational
procedure prior to such calculation is explained briefly below.
A. Computational procedure
The nuclear configuration of the single molecule is taken from an X-ray
diffraction data of the unit cell of the monoclinic geometry at 223 K. The entire
nuclear dimension is rotated and translated in order to align two carbon nuclei,
30
attached to silicon atom, on the spinning axis. The general mathematical procedure
is explained here.
Let the original X-ray coordinate frame be denoted by X Y Z, the coordinate
frame after the rotation Rz(Φ) by x’y’z’ and the coordinate frame after the rotation
RN(θ) by x’’y’’z’’ [34]. Specifically,
(12)
(13)
At first, the angle Φ is calculated in order to make the identical y’
component of the two carbon nuclei to be aligned to spinning axis, then the new
frame of x’y’z’ is obtained by applying equation (12). After that the angle θ is
calculated in order to make the identical x’’ component of these two carbon atoms
and it is followed by the frame x’’y’’z’’ after applying equation (13). The final step
is the translation of the x’’ and y’’ components to align these carbon nuclei on the
spinning axis (z axis).
During the calculations, only phenylene part with eight atoms [four
Hydrogen and four Carbon atoms] is rotated through the spinning axis, on the basis
of equation (12), at different angles so that geometry of the rest of the atoms
remains identical and single point energy calculations are performed at each
rotation.
31
B. Rotational Potential Energy Surface
The single point energy calculation of the isolated gyroscope is carried out
with HF/STO-3G, HF/6-31G (d,p) and B3LYP/6-31G (d,p) level of calculations.
The energy in wave number (cm-1) is plotted against the angle of rotation as shown
in fig 7.It shows that the general pattern of the rotational potential energy surface is
repeated in all three levels of calculations. All three surfaces, up to the first half,
have two local minima at around 1.8 and 3.0 radian and a global minimum at
around 0.5 radian. The surface calculated from HF/STO-3G level theory produces
an energy barrier of about 1700 cm-1, that from HF/6-31G (d,p) produces about
1600 cm-1 and B3LYP/6-31G (d,p) produces about 1100 cm-1.
1600
1400
1200
1000
800
600
400
200
0
Pot
entia
l ene
rgy
(cm
-1)
2.01.51.00.50.0Angle of rotation ( rad.)
blue line: HF/ STO-3Gred line: HF/ 6-31G (d,p)black line:B3LYP/ 6-31G (dp)
By comparing these two processes of Gaussian explained in section 2.1.1
and 2.1.2, it is concluded that the rotational potential energy surface derived by
fully optimizing the X-ray structure seems very far from that derived from the
single point energy calculation. Thus by considering the fact that “single point
energy calculations with an X-ray nuclear frame give an adequate approximation to
32
Fig. 7 Rotational PES from SP calculation
the best theoretical and experimental values” [33] , the single point energy
calculation process of Gaussian approaches more towards the experiments than the
fully optimized Gaussian process.
However, quantum chemistry calculation with full optimization produces
several interesting properties. Geometry optimizations usually attempt to locate
minima on the potential energy surface, thereby predicting equilibrium structures
of molecular systems. Optimization can also locate transition structures as well as
ground state structures, since both correspond to stationary points on the potential
energy surface. Thus, potential energy surface with full optimization at each point
adds enough strength towards the ab-initio molecular dynamics than single point
energy calculation.
2.1.3 DFTB approaches: Full optimization case
Since it is known that the DFTB is very feasible and reliable technique of
computational calculation, the accuracy of this method is evaluated by optimizing
the isolated siloxaalkane gyroscope.
The X-ray geometry is optimized by using a “conjugate gradient algorithm”
[35, 36] in DFTB. The conjugate gradient method is an iterative method, so it can
be applied to sparse systems that are too large to be handled by direct methods.
Thus it is also used to solve unconstrained optimization problems such as energy
minimization.
The following tables show the comparison between Si-O-Si bond angles in
each arm and the distance between Oxygen atoms of each arm in X-ray and DFTB-
optimized structure of the Siloxaalkane gyroscope.
Table 4: Variations in the Si-O-Si bond angles
33
Structure (isolated molecule) Si-O-Si bond angle in each arm (degree)X-ray 169.4 172.6 162.9Optimized[DFTB] 135.7 135.5 134.6
Table 5: Variations in the distance between Oxygen atoms of each arm
Structure (isolated molecule) Distance between Oxygen atoms of each arm (A˚)X-ray 8.99 8.67 8.50Optimized [DFTB] 10.20 9.69 9.24
The great variation in Si-O-Si bond angle and the position of the
siloxaalkane arms, in Hartree-Fock calculation level, is the main reason of
deviation of the optimized structure from the X-ray. In reference to it, the DFTB-
optimized structure of an isolated siloxaalkane molecule is also strongly deviated
from the X-ray structure. It is also observed that the siloxaalkane arms undergo
strong expansion and try to keep them away from the phenylene rotor during
optimization. However, the structure optimized by DFTB scheme seems identical
to that optimized by ab initio methods.
Thus it is concluded that consideration of the periodic boundary condition
during optimization is necessary to reproduce the structure. Similarly it is also
noted here that the single point energy calculation of DFTB with an X-ray structure
as an input is also useful to compare with the similar calculation under Gaussian-
03 package [32].
2.1.4 DFTB approaches: Static calculation case
Just like the single point calculation of the Gaussian processes, DFTB also
has the similar advantage. Since the key word “driver” in DFTB is responsible for
changing the geometry of the input structure during the calculation, “Driver {}”
terminology enforces the DFTB scheme for the single point energy calculation
34
with the input geometry. In DFTB language, it is termed as a static calculation
[37].
A. Computational procedure
The general computational procedure prior to the DFTB static calculation is
explained in subsection A of section 2.1.2. At each angle of phenylene rotation, the
static calculation is carried out in order to compare the effect of the static stator
with relaxed one.
B. Rotational Potential Energy Surface
In order to check the validity of the “DFTB” method in reference to the
siloxaalkane rotor molecule, the Gaussian approach of single point energy
calculation is one of the choices. As the mentioned method is based on the Density
Functional Theory (DFT), conventional DFT of Gaussian approach has become
our selective.
The potential energies from single point calculation of DFT and DFTB are
plotted against angles of rotation. Fig. 8 shown below is the rotational potential
energy surface of the phenylene at these two levels of calculations. The upper
surface is derived from B3LYP/6-31G (d,p) and the lower surface is from the
DFTB calculation.
It is clear that the phenylene group has a rotational barrier of about 1100 cm -
1in B3LYP/6-31G (d,p) level of calculation whereas the barrier is reduced to
around 900 cm-1 in DFTB level.
It is found that there are no any parameters, with respect to phenylene rotor,
which directly explain this appearance of the potential energy surface. Thus it is
35
expected that the mentioned barrier is due to the Vanderwaal’s and dipole- dipole
interaction between the atoms of the phenylene rotor and the siloxoalkane stator.
1000
800
600
400
200
0
Pot
entia
l Ene
rgy
(cm
-1)
2.01.51.00.50.0Angle of rotation ( rad.)
blue line: B3LYP/ 6-31G(d,p)red line: DFTB
By comparing these energy surfaces derived from two different levels of
calculations, one can conclude that the stable and intermediate positions of the
phenylene rotor are appeared at the identical angle of rotation. This reproduction of
the main feature of the potential energy surface by the DFTB method verifies the
validity of it to probe further.
2.2 Dynamics of the Molecular Gyroscope under crystal conditions
According to the crystallography, the crystal of the siloxaalkane rotor
molecule is classified as a molecular crystal. In such a crystal, the constituent
particles are molecules which are formed by covalent bonds between the atoms.
The molecules are held together by weak physical bonding such as vanderwaal’s
forces or dipole-dipole interactions.
36
Fig. 8 Potential energy as a function of the angle of rotation.
Although the forces between molecules in crystals are weak and short-range,
and the overlap between the orbital of adjacent molecules in the lattice is small,
there are substantial differences in the several electronic properties of crystals and
free molecules [38]. Some of these differences arise from the interactions between
the electronic states of a molecule and those of molecules in the immediate
vicinity, while others arise as a consequence of the collective properties of the
crystal lattice. Similarly, there are differences between the optical properties of
solids and free molecules, of which some may be regarded as effects resulting from
changes in the local environment of a molecule or group while others are
characteristic of the lattice as a whole [39].
This sensitivity of the electronic properties to the structure of, and
interactions within, molecular crystals implies that studies of the dynamics of the
molecule under crystalline condition can yield a detail dynamics.
2.2.1 Gaussian approaches
Gaussian-03 package [32] has an advantage of calculating electronic
properties under Periodic Boundary Conditions [PBC]. Wang et. al has reported
that the results obtained by considering the unit cell of Single Wall Carbon Nano
Tubes [SWNTs] with 20 carbon atoms for single circumference are consistent with
the experimental data [40]. Although there has been much success in applying
these methods to larger systems, they are found to be too slow for the investigation
of many interesting problems. Moreover, these methods hardly work for the huge
molecular crystal system with Vanderwaal’s force of interactions as a lattice force.
Since calculating rotational potential energy surface under crystal condition
is the key point of this work to seek the dynamics of phenylene rotation. On such
sense, considering periodic boundary condition with Gaussian level of calculation
37
is found to be troublesome and almost impossible. None of the Gaussian jobs were
found to be terminated normally.
2.2.2 DFTB approaches: Static calculation
In order to understand the effect of the molecules in the immediate vicinity
during single point calculation, the DFTB-static calculation under periodic
boundary condition is performed. Since the nature of the rotational potential
energy surface of the isolated molecule derived by using same calculation is
already presented in sub-section B of section 2.1.4, it is interesting to compare the
result with and without periodic boundary condition.
A. Computational procedure
The nuclear configuration of one unit cell is taken from an X-ray diffraction
data at 223 K. The entire nuclear frame of the system is rotated and translated as
usual. This operation is followed by the rotation of the eight atoms (excluding two
axial Carbon atoms ) of the phenylene rotor at different angles on the spinning
axis by using equation (12) and then the entire nuclear configuration is brought
back to normal form to recover the unit cell structure.
B. Rotational Potential Energy Surface
Fig. 9 shown below illustrated the nature of the rotational potential energy
surface with and without periodic boundary condition. It can be said that with and
without periodic boundary condition under DFTB-static calculation, the general
feature of the rotational potential energy surface is identical. The rotational
potential energy barrier of the phenylene rotor also seems equivalent.
Thus it can be concluded that the effect of the neighboring molecule seems nil.
Therefore, in this case, considering periodic boundary condition during single
38
point energy calculation of DFTB does not produce any extra information.
However, it is well known that the significant properties of the isolated molecule
are too far from its crystal. Such deviations on the properties are due to the
negligence of the surrounding molecular array.
800
600
400
200
0
Pot
entia
l Ene
rgy
(cm
-1)
2.01.51.00.50.0Angle of rotation ( rad.)
blue line: PBC conditionred line: Isolated molecule
.
Above explanation makes it clear that “DFTB-static calculation” with
periodic boundary condition is not efficient at all for explaining the effect of the
surrounding molecules during the phenylene rotation. Therefore the only option we
have here is the “DFTB-Optimization calculation” under periodic boundary
condition. The general findings of it are presented in the following section.
2.2.3 DFTB approaches: Full optimization case
Instead of fixing the nuclei during periodic boundary condition calculations
as in “DFTB-static calculation”, relaxing all the nuclei is expected to produce some
changes on the potential energy surface. The X-ray geometry of the unit cell is
optimized, in the presence of periodic boundary condition, by using a “conjugate
gradient algorithm” [35, 36] in DFTB. It is pointed out that DFTB reproduced
39
Fig. 9 Potential energy surface with and without periodic boundary condition
almost similar structure to that of an X-ray in the unit cell. For making it clear,
these structures are shown in fig. 10 and 11.
Even though, the optimized structure seems identical to that of the X-ray structure in several aspects; the Si-O-Si bond angle in each siloxaalkane arm is still deviating, as mentioned in table 4.
Table 4: Variations in the Si-O-Si bond angles
Structure Si-O-Si bond angle in each arm (degree)X-ray 169.4 172.6 162.9Optimized [DFTB] [Isolated molecule]
135.7 135.5 134.6
Optimized [DFTB][Periodic boundary condition]
121.1 119.99 121.6
Moreover it is also observed that the degree of deviation appeared on the
“DFTB-optimized structure of the isolated molecule” is strongly reduced in the
“DFTB-optimized structure with periodic boundary condition”. The optimized
structure of the unit cell of the molecular crystal has the siloxaalkane arms with
almost at the identical positions unlike their positions in the optimized structure of
the isolated molecule case. This is also clarified by the closeness, of the distance
between Oxygen atoms of each arm, with that of the X-ray structure, mentioned in
table 5. This is why; one can confirm that periodic boundary condition is the
mandatory for getting real molecular dynamics.
40
41
Fig. 10 X-ray geometry at 223 K
Fig. 11 DFTB optimized geometry
Table 5: Variations in the distance between Oxygen atoms of each arm.
Structure Distance between Oxygen atoms of each arm (A˚)X-ray 8.99 8.67 8.50Optimized [DFTB] [Isolated molecule]
10.35 9.78 9.55
Optimized [DFTB][Periodic boundary condition]
9.684 8.688 8.662
A. computational procedure
The procedure of making nuclear frame of the molecular system is similar to
that explained in subsection A of section 2.2.2. During “DFTB- optimization
calculation”, all the atoms excluding four spatial carbon atoms of phenylene are
relaxed at each angle to assure the consistency on the angle of phenylene rotation.
B. Rotational Potential energy surface
The potential energies obtained from “DFTB-optimization” calculation are
plotted against angles of rotation. Fig. 12 shown below is the rotational potential
energy surface of the single siloxaalkane gyroscope under crystal condition. The
upper surface represents the surface derived from the DFTB-static calculation and
the lower surface represents the surface derived from the full optimization under
DFTB with periodic boundary condition.
The potential energy surface makes it clear that the phenylene group has a
rotational barrier of about 900 cm-1 in “DFTB-Static” calculation whereas the
barrier is reduced to around 450 cm-1 when the structure is fully optimized in
DFTB level.
42
The initial angle of phenylene rotation in X-axis indicates the X-ray
structure. The potential energy of around 250 cm-1corresponding to this structure
indicates one of the local minima. It illustrates that the X-ray structure does not
represent the most stable structure. The appearance of one of the global minima at
around 0.5π radian indicates the stable structure of the molecule under crystal
condition.
800
600
400
200
0
Pot
entia
l Ene
rgy
(cm
-1)
2.01.51.00.5Angle of rotation ( rad.)
red line:DFTB -staticblue line: DFTB -relaxed
.
The geometry of a molecule and the static and dynamic calculations
determine many of its physical and chemical properties. In computational
chemistry, people are specifically concerned with optimizing: bond angles
(degrees), bond distances (angstroms) and dihedral angles (degrees). Thus in this
case, optimizing the siloxaalkane spokes or stator part of the siloxaalkane rotor
makes the great variation of potential energy. Therefore it can be summarized that
motion of the stator plays an important role to describe the rotor–stator interaction.
Since the rotational energy barrier of the isolated siloxaalkane rotor,
calculated under the full optimization of the B3LYP/6-31G level, presented in
43
Fig. 12 Potential energy as a function of the angle of rotation
section 2.1.1, shows an energy barrier of about 100 cm-1. Whereas the barrier
becomes around 450 cm-1 under the “DFTB-full optimization” case with periodic
boundary conditions. This comparison indicates that the neighboring array of the
molecules reduce the rate of phenylene rotation by increasing the energy barrier of
about four times. Thus, consideration of the molecular crystal during
computational calculation is very necessary to understand the crystalline effect.
2.3 DFTB Molecular Dynamics (MD) Simulation
Simulations of the materials tell us in which way the building blocks interact
with one another and with environment, determine the internal structure, the
dynamic processes and the response to the external factors such as temperature,
Pressure, electric and magnetic field, etc. For being a complement and alternative
to an experimental research, a fast and efficient simulation method that produces
the results in good agreement with experiments as well as ab initio calculations
based on DFT and HF theory is needful.
DFTB MD simulation with velocity verlet algorithm is one of the
recommended methods that fulfill the criteria mentioned above. Thus in this case
too, the velocity and the position of the particle with in small time increment are
computed by using this algorithm under DFTB program package [35, 37].
2.3.1 Rotary motion of the phenylene group
The rotary motion of the phenylene rotor is studied under the conditions of
low and high temperarure. The results obtained are explained below.
A. Low temperature case
The angle of phenylene rotation is plotted against the sampling time to
explain the dynamics of the rotor in fig 13. The average Kinetic temperature and
44
sampling time are about 600 K and 14 ps respectively under the condition of NVE
simulation with periodic boundary condition.
As mentioned earlier, 0.5π and -0.5π radian represents the two most stable
positions of the siloxaalkane molecule. These are indicated by the dotted lines in
the corresponding figure. The initial condition of the siloxaalkane rotor represents
an X-ray structure. This figure also shows that the structure at the angle of rotation
about Φ = 0 represents one of the local minima.
-1.0
-0.5
0.0
0.5
1.0
(
rad)
121086420
Time (ps)
When the rotating time increases up to about 500 fs, the rotational trajectory
reaches to one of the stable positions. In other words, the siloxaalkane rotor
reaches to one of the most stable positions at around Φ = −0.5 radian with in 500
fs. Then it stays on that stable state for about 14 ps even though the molecule has
enough energy to overcome the potential energy barriers.
45
Fig.13 Time course of the rotor at around 600 K
B. High temperature case
The angle of phenylene rotation is plotted against the sampling time to
explain the dynamics of the siloxaalkane rotor at high temperature in fig. (14). The
average Kinetic temperature and sampling time are ~1200 K and 42 ps respectively
under the condition of NVE simulation with periodic boundary condition.
-1.0
-0.5
0.0
0.5
1.0
(
rad)
403020100
Time (ps)
In this figure too, 0.5π and -0.5π radian indicated by two dotted lines,
represents the two most stable positions of the siloxaalkane molecule. The initial
condition of the molecule represents an X-ray structure. The figure shows that the
corresponding structure with angle of rotation about Φ = 0 represents one of the
local minima. The figure shows that, in average, the phenylene rotor flips from one
stable position to another in 20 ps. The flipping motion of this gyroscope is
observed for the first time under the condition of high temperature ~1200 K.
46
Fig. 14 Time course of the rotor at around 1200
K
2.4. Summary
The dynamics of the phenylene rotation in siloxaalkane molecular
gyroscope is studied under the DFTB and Gaussian approaches. The
Gaussian scheme is applied only to the isolated molecular case whereas
DFTB scheme is applied to the isolated as well as the crystalline condition.
The theme of these general calculations is summarized below:
The X-ray structure of the siloxaalkane molecule with phenylene rotor
encased in three – spoke silicon-based stator is reproduced by an ab initio
quantum chemistry calculation.
The validity of the DFTB method is checked in reference to this molecular
gyroscope and it gives qualitatively the same result as of DFT methods.
Eventhough, the DFTB-optimized structure of the isolated molecule is
seemed to be similar; the nature of the potential energy surface is very
asymmetric and unsmooth as well as dissimilar to that of ab initio Gaussian
calculations based on Hartree-Fock and B3LYP methods. Thus, DFTB-full
optimization case is not recommended method for the isolated siloxaalkane
molecule.
All the calculations based on the Hartree-Fock and B3LYP under periodic
boundary conditions are found to be unable to interpret the vanderwaal’s and
dipole-dipole interaction exists as a lattice force in molecular crystal.
However, DFTB package is found to work well with low computational cost
and high efficiency.
The rotational energy barrier of the isolated molecular gyroscope, calculated
by B3LYP, is observed to almost four times less than that under crystal
47
condition, calculated by DFTB scheme. Thus the phenylene rotor is found to
interact strongly with the periodic molecular array during the rotation.
Moreover, the MD Simulations of the siloxoalkane molecular crystal
under DFTB package is also carried out with the velocity verlet algorithm.
Following conclusions are drawn based on the results obtained.
DFTB simulation is found to produce the stable structures at the same angle
of phenylene rotation to that observed in the potential energy surface derived
by “DFTB-full optimization calculation” with periodic boundary condition.
Flipping motion of the phenylene group as a rotor is observed and the
dynamics of the phenylene rotor is found to be strongly depending on the
siloxaalkane stator and the neighboring molecular array. It speculated us that
crystalline lattices should require systems with molecular rotors that
experience steric contacts that constitute each other’s main rotational barrier.
In low temperature case, the phenylene rotor is found to stay at the stable
position at least 1ns However, at high temperature of about 1200K, the
phenylene rotor undergoes flipping in an average time of 20ps Thus this
flipping verifies the X-ray observation of the facile phenylene rotation in
solid state [10].
Though the limits and the strength of the DFTB model in reference to this
molecular gyroscope are summarized above; our findings from DFTB calculation
are comparable to that of the ab initio calculations based on Hartree-Fock and
B3LYP methods.
Setaka et al. has mentioned that synthesis and analysis of the siloxaalkane
spokes is relatively easy and by using flexible siloxaalkane side chains, phenylene
48
rotation of this molecular gyroscope is also expected to be temperature-controlled
[10]. In addition, optical control of the rotation may be feasible by introducing the
polar substituents on the phenylene rotor of this molecular gyroscope.
Molecular rotors should provide a useful tool in creating nanometer scale
machines and phenomena. The ability to characterize the dynamics of well
organized, three-dimensional arrays of these molecules is shown in this booklet.
We believe that there is much to be learned in the construction and analysis of
crystalline solids with structurally programmed motions. We also expect that
phenylene flipping in molecular rotors with intramolecular steric shielding will
increase the rate of phenylene rotation and will help the synthetic chemists to
achieve the preparation of fast molecular compasses and molecular gyroscopes.
49
Bibliography
1. Steven D. K., Peter D. J., Rosa S. and Garcia-Garibay M. A. Phys. Rev. B,
2006 74, 054306.
2. Zaira D., Hung D., M. Jane S. and Garcia-Garibay M. A. J. Am. Chem. Soc.
2002, 124, 7719.
3. Dominguez Z., Dang H., Strouse M. J. and Garcia-Garibay M. A. J. Am.
Chem.Soc. 2001, 124, 2398.
4. Dominik H. and Josef M. PNAS, 2005, 102, 14175.
5. For some general text about the molecular machine, please refer: Molecular
Machines and Motors, Vol. Editor: Sauvage J.P. Springer pub.
6. Kuimova K.M., Gokhan Y., James A. Levitt and Klaus S., J. Am. Chem. Soc.
2008, 130, 6672.
7. Hoki K., Yamaki M., Koseki S. and Fujimura Y. J. Chem. Phys. 2003,118,
497.
8. Astumian, R. D. Science 1997, 276, 917.
9. Dominguez Z., Dang H., Strouse M. J. and Garcia-Garibay M. A., J. Am.
Chem. Soc. 2002, 124, 2398.
10.Setaka W., Ohmizu S., Kabuto C. and Kira M., Chem. Lett. 2007, 36, 1076.
11.Silverstein R., Bassler M. and Morrill T. C. Spectrometric Identification of
Organic Compounds 1991, Wiley Pub.
12.Elsa C., Gerald S R., Eve T.and Serge A., "Precise and accurate quantitative 13C NMR with reduced experimental time", 2007 Talanta 71, 1016.
13.Otero J. G., Porto M., Rivas J. and Bunde A. Phys. Rev. Lett. 84.
14.Lambert D.E. and Wilson M.A. CSIRO Division of Fossil Fuels, Australia.
15.Pines A., Gibby M. G. and Waugh, J. S. J. Chem. Phys. 1973, 59, 569.
50
16.For some general reviews of the CP/MAS experiment, please refer: (a)
Schaefer J. and Stejskal E. O. Top. Carbon-13 NMR Spectrosc. 1979, 3, 283.
(b) Fyfe C. A. Solid State NMR for Chemists; CFC Press: Guelph, Ontario,
1983. (c) Yannoni, C. S. Acc. Chem. Res. 1982, 15, 201-208.
17.Lawrence B. A., David M. G., Terry D. A. and Ronald J.P.
J.Am.Chem.Soc.1983, 105, 6697.
18.S Opella J. and Frey M. H., J. Am. Chem. Soc. 1979, 101, 5854.
19.Xiaoling W, Shanmin Z and Xuewen W, J. Magn. Reson. 1988, 77, 343
20.(a) Porezag D. et al. Phys. Rev. B 1995, 51, 12947.
(b) Seifert G. et al. Int. J. Quant. Chem. 1996, 58, 185.
21.Kohn W. and Sham L. J., Phys. Rev. A 1965, 140, 1133.
22. D. Porezag T. F. and Kohler T. Phys. Rev.B 1995, 51, 19.
23. Seifert G. J. phys.chem.A 2007, 111, 5609.
24.Alder, B. J. and Wainwright T. E. 1959 J. Chem. Phys. 31, 459.
25. Rahman A. 1964, Phys Rev 136, 405.
26. Bernal, J.D. 1964, Proc. R. Soc. 280, 299.
27. For some general text about the computational applicatiions, please refer: (a)
Schlick T. (1996). "Pursuing Laplace's Vision on Modern Computers". (b)
Mesirov J. P., Schulten K. and Sumners D. W. “Mathematical Applications to
Biomolecular Structure and Dynamics”, IMA Volumes in Mathematics and Its
Applications. Springer pub. ISBN 978-0387948386.
28.For MD simulation note, please refer: http://www.ch.embnet.org/MD_tutorial
29. Verlet L., Phys. Rev. 1967,159, 98.
30. Allen P. M., NIC Series, Vol. 23, ISBN 3-00-012641-4, pp. 1-28, 2004. “John
Von Neumann Institute for computing” pub.
31.“Exploring Chemistry with Electronic structure methods” by Jamaes B.
Foresman and Eleen Frisch, Gaussian Inc.
51
32.Gaussian 03, second edition, User’s reference.
33.J. Garcia E. and Corchado J. C., J. Phys. Chem. 1995,99, 8613.
34.ANGULAR MOMENTUM: “UNDERSTANDING SPATIAL ASPECTS IN
CHEMISTRY AND PHYSICS” BY RICHARD N. ZARE, Wiley-
Interscience Pub.
35.B. Aradi et al. J. Phys. Chem. A, 111, 2007, 5678.
36.“An Introduction to the Conjugate Gradient Method”, School of Computer
Science Carnegie Mellon University Pub.
37.DFTB+ Snapshot 081217 “USER MANUAL”.
38.“Molecular crystals”, second edition by J.D WRIGHT, Cambridge University
press. ISBN 0-521-47730-1.
39.Organic Molecular solids: “Properties and applications” Edited by William
Jones. ISBN 0-8493-9428-790000.
40.Wang H.W., Wang B.C., Chen W.H. and Hayashi M., J. Phys. Chem. A, 2008,
112, 1783.
52
Appendix I
DFTB code of an isolated molecule for the static calculation
ParserOptions = { ParserVersion = 3 }
Driver = {}
Hamiltonian = DFTB {
SlaterKosterFiles = {
C-C = "C-C.skf"
C-H = "C-H.skf"
H-C = "C-H.skf"
H-H = "H-H.skf"
C-O = "C-O.skf"
O-C = "C-O.skf"
O-H = "H-O.skf"
H-O = "H-O.skf"
O-O = "O-O.skf"
C-Si = "C-Si.skf"
Si-C = "C-Si.skf"
H-Si = "H-Si.skf"
Si-H = "H-Si.skf"
53
O-Si = "O-Si.skf"
Si-O = "O-Si.skf"
Si-Si = "Si-Si.skf"
}
MaxAngularMomentum = {
H = "s"
C = "p"
O = "p"
Si = "d"
}
}
Geometry = GenFormat {
195 C
H C O Si
188 2 7.57623500 4.21290000 10.23256400
190 2 8.94693500 4.20141400 10.30338600
192 2 9.01531200 4.43833400 7.94993100
194 2 7.64143200 4.42991000 7.90029600
189 1 7.02259341 4.11261753 11.17948606
54
191 1 9.42161308 4.09855057 11.28775705
193 1 9.56459229 4.52744131 7.00128372
195 1 7.16368341 4.50875039 6.91488866
18 2 6.85158509 4.33239427 9.04435384
19 2 9.73203029 4.34636055 9.15166401
1 4 4.98417619 4.38569806 8.97752282
2 4 11.59667208 4.47909048 9.21003085
3 4 4.17165635 7.84200345 11.87019509
4 4 8.11608005 9.82443654 10.75394431
5 4 8.51201847 10.49071432 8.08748665
6 4 12.34358187 8.30341611 6.75864319
7 4 4.20309946 0.20012707 10.63824697
8 4 7.94071613 0.92907190 13.17973496
9 4 8.32336160 3.02967658 14.97558367
10 4 12.38853446 4.82611942 13.72109327
11 4 4.16867634 5.22978751 4.53711835
12 4 8.12696310 3.28159166 3.16488741
13 4 8.64618387 1.15795174 4.89806207
14 4 12.65113099 0.58354118 7.05758990
55
15 3 7.60831089 9.84113344 9.23652869
16 3 7.55259340 2.38093761 13.73011926
17 3 9.00218762 2.60993190 4.32591072
20 2 4.43375906 6.04441359 9.66184645
21 1 5.06160159 6.82520739 9.20477369
22 1 3.40102472 6.23299188 9.33072290
23 2 4.51356105 6.12847093 11.18335050
24 1 5.51632736 5.82687077 11.52465306
25 1 3.80145168 5.41867667 11.63163247
26 2 5.45758112 9.03827068 11.21296554
27 1 5.36829902 9.07074600 10.11521403
28 1 5.21445183 10.04966638 11.57547986
29 2 6.89169048 8.68235492 11.60103280
30 1 7.11153539 7.64356177 11.30742478
31 1 7.00988284 8.73351984 12.69432685
32 2 9.71528681 9.21861097 7.38980463
33 1 9.47523209 8.24424859 7.84464259
34 1 9.51826588 9.10811900 6.31106905
35 2 11.19633710 9.52658755 7.60307710
56
36 1 11.42782120 9.54482732 8.67884197
37 1 11.43152214 10.52998455 7.21356876
38 2 11.92200377 6.53998691 7.25250044
39 1 10.88410929 6.34939574 6.93794859
40 1 12.55828554 5.85986913 6.66467675
41 2 12.07746722 6.23505119 8.73943578
42 1 11.46460923 6.93134808 9.33307340
43 1 13.12355490 6.39289212 9.04461755
44 2 4.32360768 2.99340928 10.04598045
45 1 4.84102905 3.04939208 11.01679364
46 1 3.25677345 3.17440525 10.24570050
47 2 4.50374816 1.60762041 9.43086254
48 1 5.52906296 1.49713749 9.04298151
49 1 3.82901361 1.49621810 8.56954353
50 2 5.35267228 0.42409129 12.10575599
51 1 4.99226003 1.27539052 12.70471056
52 1 5.27664151 -0.46641064 12.74920981
53 2 6.80916934 0.65262213 11.70719318
54 1 6.88275682 1.53301832 11.04884524
57
55 1 7.17258603 -0.20607098 11.12138055
56 2 9.66473045 4.21873700 14.39086131
57 1 9.39664757 4.55591716 13.37718050
58 1 9.63693011 5.11642040 15.02854062
59 2 11.08291148 3.65026772 14.38683733
60 1 11.12055633 2.73170925 13.78284206
61 1 11.37151085 3.36441172 15.41088465
62 2 12.02496668 5.25319949 11.92685984
63 1 11.00196848 5.65830318 11.88004451
64 1 12.69666089 6.07043906 11.62378059
65 2 12.17533238 4.08607484 10.95215014
66 1 11.60826739 3.21304970 11.31244397
67 1 13.23078754 3.77700222 10.90871390
68 2 4.44540253 4.18667125 7.19154565
69 1 5.07644975 3.40998051 6.73154329
70 1 3.41649165 3.79738922 7.18326704
71 2 4.50816599 5.47238238 6.36960063
72 1 5.49948333 5.94133848 6.46850826
73 1 3.77987471 6.19741348 6.76409878
58
74 2 5.39428863 4.00836709 3.80274643
75 1 5.27847267 3.04689239 4.32649740
76 1 5.11047425 3.82941790 2.75355904
77 2 6.84804952 4.47371153 3.87280210
78 1 7.12093023 4.67519763 4.91995234
79 1 6.94375669 5.43598186 3.34460145
80 2 9.89651457 0.86467993 6.26964838
81 1 9.81393471 1.68738783 6.99780749
82 1 9.64365008 -0.06032503 6.81032953
83 2 11.32673787 0.78093201 5.73891827
84 1 11.55926103 1.69081134 5.16271356
85 1 11.41275943 -0.06247695 5.03570215
86 2 12.33953758 1.82971770 8.42820156
87 1 11.38359297 1.55861766 8.90414573
88 1 13.11632305 1.72116069 9.19986135
89 2 12.29363302 3.27792614 7.94868619
90 1 11.67162066 3.35902489 7.04321507
91 1 13.30414898 3.60762228 7.66602872
92 2 4.29543703 7.76876771 13.72843299
59
93 1 5.28888567 7.42931308 14.02777438
94 1 3.55857195 7.07472207 14.13641384
95 1 4.12265439 8.75304289 14.16748684
96 2 2.48795775 8.44565121 11.34517372
97 1 2.45110083 9.53626087 11.36422299
98 1 1.71328745 8.06344315 12.01186612
99 1 2.26020901 8.11518642 10.33054833
100 2 9.83571977 9.15031872 11.02205331
101 1 10.59761707 9.88166976 10.74536961
102 1 9.99726630 8.24679177 10.43075726
103 1 9.98393896 8.89528816 12.07307244
104 2 7.98628679 11.49085872 11.59170672
105 1 8.85551224 12.12010051 11.39087039
106 1 7.90981766 11.36352985 12.67344790
107 1 7.09545668 12.02336882 11.25226992
108 2 7.33581354 10.94894045 6.71090457
109 1 6.80708812 10.06781328 6.34237540
110 1 7.86631681 11.40156269 5.87126920
111 1 6.59326090 11.66476917 7.06946791
60
112 2 9.41271825 12.04461662 8.58925114
113 1 10.18439158 11.84204503 9.33436323
114 1 8.71777781 12.77576081 9.00694749
115 1 9.89076082 12.49728866 7.71890894
116 2 14.10545567 8.73392042 7.19308696
117 1 14.36085311 8.37456937 8.19120289
118 1 14.24818466 9.81586944 7.17200877
119 1 14.79787644 8.28440263 6.47897140
120 2 12.11433183 8.44300733 4.91271736
121 1 12.83937745 7.81607680 4.39019728
122 1 12.25145342 9.47447958 4.58284836
123 1 11.11165715 8.12147798 4.62469372
124 2 2.44689887 0.18904180 11.26599099
125 1 2.35939836 -0.46777145 12.13378377
126 1 1.76084410 -0.16981981 10.49694232
127 1 2.13824818 1.19264999 11.56514378
128 2 4.58565904 -1.41377043 9.78422556
129 1 4.43795095 -2.25408836 10.46525506
130 1 5.62168382 -1.42877741 9.44060951
61
131 1 3.93627080 -1.55542084 8.91825034
132 2 9.68747139 0.79604987 12.53389207
133 1 10.41636979 0.76601047 13.34545925
134 1 9.92519503 1.64657294 11.89141679
135 1 9.80647667 -0.11458223 11.94349545
136 2 7.60304966 -0.45422171 14.39145651
137 1 8.44174727 -0.63083538 15.06710731
138 1 7.40346934 -1.38662219 13.86012754
139 1 6.72423309 -0.22047587 14.99545142
140 2 7.05146883 4.06312357 15.87365902
141 1 6.24381614 3.43482591 16.25481241
142 1 6.61422257 4.80492901 15.20236875
143 1 7.49569939 4.59159777 16.71871453
144 2 9.05729500 1.84065306 16.21097559
145 1 8.28676445 1.21030923 16.65915283
146 1 9.54603824 2.39289740 17.01585306
147 1 9.80364648 1.19314864 15.74722590
148 2 14.04903266 3.99106149 13.88194338
149 1 14.10805847 3.12033895 13.22614169
62
150 1 14.20844268 3.65721443 14.90897717
151 1 14.85550179 4.67694468 13.61668052
152 2 12.37793311 6.40277387 14.71853196
153 1 13.09313289 7.12111591 14.31335718
154 1 12.64440419 6.20196748 15.75809725
155 1 11.38771488 6.86201642 14.70073613
156 2 4.33499717 6.88292760 3.68873540
157 1 5.33001737 7.30164571 3.85083950
158 1 3.59896328 7.58825492 4.07978010
159 1 4.17548283 6.78218230 2.61344199
160 2 2.44163574 4.57854336 4.26770611
161 1 2.29301985 3.63826946 4.80106888
162 1 2.26284920 4.40201180 3.20536877
163 1 1.70095653 5.29677028 4.62473025
164 2 9.31869246 4.33132848 2.18298222
165 1 10.08721334 3.71481346 1.71239619
166 1 9.81521173 5.05720930 2.82989958
167 1 8.79506758 4.87936995 1.39863672
168 2 7.29640286 2.11538643 1.97058148
63
169 1 8.02877766 1.47870655 1.47028832
170 1 6.76415511 2.67963834 1.20208212
171 1 6.57360578 1.47452917 2.47830924
172 2 6.94175079 1.04827944 5.65299710
173 1 6.77476086 1.87465123 6.34730019
174 1 6.82389071 0.11450899 6.20583284
175 1 6.16439685 1.08533883 4.88705948
176 2 8.86932266 -0.22357268 3.65630716
177 1 9.65193575 0.02698732 2.93730267
178 1 7.95249902 -0.42678882 3.09995373
179 1 9.16162618 -1.14708874 4.16045257
180 2 14.30389025 0.83732983 6.23028072
181 1 15.10089221 0.34784760 6.79261485
182 1 14.54228132 1.89859074 6.14411891
183 1 14.28735627 0.40919081 5.22631503
184 2 12.60903771 -1.12913843 7.79659484
185 1 13.58564796 -1.38766110 8.21042385
186 1 12.35181192 -1.87347630 7.04053761
187 1 11.87420588 -1.18458470 8.60104602 }
64
For the more detail information about the geometry of the molecule etc, the
reader is requested to contact the concerned person at the Mathematical Chemistry
Laboratory [KONO LAB], Department of Chemistry, Aoba-yama Campus,
Tohoku University, Sendai.
65