Ab Initio Dynamics Simulation of the Molecular Gyroscope

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修修修修 Ab Initio Dynamics Simulation of the Molecular Gyroscope (修 修 修修修修修修修修Anant Babu Marahatta 21 Graduate school of Science Department of Chemistry 1

description

A MS thesis by Anant Babu Marahatta, Department of chemistry, Tohoku University, Japan

Transcript of Ab Initio Dynamics Simulation of the Molecular Gyroscope

Page 1: Ab Initio Dynamics Simulation  of the Molecular Gyroscope

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

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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

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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

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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

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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

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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.

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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

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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

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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

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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.

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Fig. 3 An X- ray structure of Siloxaalkane rotor. Gyroscopic parts are labeled.Hydrogen atoms are omitted for clarity.

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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

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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.

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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

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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

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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

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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

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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.

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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.

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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

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.

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.

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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.

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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,

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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,

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(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.

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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

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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

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Page 25: Ab Initio Dynamics Simulation  of the Molecular Gyroscope

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

Page 26: Ab Initio Dynamics Simulation  of the Molecular Gyroscope

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

Page 27: Ab Initio Dynamics Simulation  of the Molecular Gyroscope

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

Page 28: Ab Initio Dynamics Simulation  of the Molecular Gyroscope

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

Page 29: Ab Initio Dynamics Simulation  of the Molecular Gyroscope

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

Page 30: Ab Initio Dynamics Simulation  of the Molecular Gyroscope

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

Page 31: Ab Initio Dynamics Simulation  of the Molecular Gyroscope

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

Page 32: Ab Initio Dynamics Simulation  of the Molecular Gyroscope

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

Page 33: Ab Initio Dynamics Simulation  of the Molecular Gyroscope

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

Page 34: Ab Initio Dynamics Simulation  of the Molecular Gyroscope

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

Page 35: Ab Initio Dynamics Simulation  of the Molecular Gyroscope

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

Page 36: Ab Initio Dynamics Simulation  of the Molecular Gyroscope

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.

Page 37: Ab Initio Dynamics Simulation  of the Molecular Gyroscope

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

Page 38: Ab Initio Dynamics Simulation  of the Molecular Gyroscope

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

Page 39: Ab Initio Dynamics Simulation  of the Molecular Gyroscope

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

Page 40: Ab Initio Dynamics Simulation  of the Molecular Gyroscope

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

Page 41: Ab Initio Dynamics Simulation  of the Molecular Gyroscope

41

Fig. 10 X-ray geometry at 223 K

Fig. 11 DFTB optimized geometry

Page 42: Ab Initio Dynamics Simulation  of the Molecular Gyroscope

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

Page 43: Ab Initio Dynamics Simulation  of the Molecular Gyroscope

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

Page 44: Ab Initio Dynamics Simulation  of the Molecular Gyroscope

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

Page 45: Ab Initio Dynamics Simulation  of the Molecular Gyroscope

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

Page 46: Ab Initio Dynamics Simulation  of the Molecular Gyroscope

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

Page 47: Ab Initio Dynamics Simulation  of the Molecular Gyroscope

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

Page 48: Ab Initio Dynamics Simulation  of the Molecular Gyroscope

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

Page 49: Ab Initio Dynamics Simulation  of the Molecular Gyroscope

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

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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

Page 51: Ab Initio Dynamics Simulation  of the Molecular Gyroscope

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

Page 52: Ab Initio Dynamics Simulation  of the Molecular Gyroscope

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

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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

Page 54: Ab Initio Dynamics Simulation  of the Molecular Gyroscope

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

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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

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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

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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

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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

Page 59: Ab Initio Dynamics Simulation  of the Molecular Gyroscope

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

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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

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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

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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

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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

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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 }

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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