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PossibleGroupsForMolecules.lyx
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PossibleGroupsForMolecules.lyx
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#LyX 2.3 created this file. For more info see http://www.lyx.org/
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\begin_layout Title
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What are the possible axial point groups for a
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\begin_layout Title
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collection
\end_layout
\end_inset
system of atoms with a certain size or formula?
\end_layout
\begin_layout Author
Nike Dattani
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www.hpqc.org
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nike@hpqc.org
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\begin_layout Affiliation
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HPQC Labs, Waterloo, Canada
\end_layout
\begin_layout Affiliation
\lang british
HPQC College, Waterloo, Canada
\end_layout
\begin_layout Author
Sichao Xiong
\end_layout
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sichao@hpqc.org
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HPQC Labs, Waterloo, Canada
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\begin_layout Date
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\begin_layout Abstract
For each axial point group, we determine the number of nuclei allowed, and
all possible chemical formulas.
We also present a set of principles that can be used for proofs about the
possibility or impossibility for a certain point group to apply to a molecule
of a certain size and/or chemical formula.
For tri-nuclear and tetra-nuclear molecules, we provide examples of molecules
with each possible chemical formula if such molecules are known, and hence
we determine the types of molecules with under five nuclei that are still
yet-to-be discovered (in terms of point groups).
We also resolve some inconsistencies in a table by P.
Atkins, M.
Child and C.
Phillips
\begin_inset Note Note
status open
\begin_layout Plain Layout
found in
\begin_inset CommandInset citation
LatexCommand citep
key "1970Atkins"
literal "false"
\end_inset
\end_layout
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, add missing entries to it, and improve it further in a number of ways.
\begin_inset Note Note
status open
\begin_layout Plain Layout
The paper assumes that you know Schoenflies notation, and the elements in
each group, and what they mean? Or we could give the elements in each group
and just expect them to know what sigma_h, etc.
mean? Or we can explain those too?
\end_layout
\end_inset
\end_layout
\begin_layout Section
\lang british
\begin_inset CommandInset label
LatexCommand label
name "sec:Introduction"
\end_inset
Introduction
\end_layout
\begin_layout Standard
\lang british
One might wonder why, despite group theory being applied to molecules for
about 100 years
\begin_inset CommandInset citation
LatexCommand citep
key "1929Bethe"
literal "false"
\end_inset
, the questions that are being addressed in this paper do not appear to
have been answered yet.
Indeed we have asked all living authors of
\begin_inset CommandInset citation
LatexCommand citep
key "1955Wilson,1963Cotton,1966Hochstrasser,hall1969group,1969Ferraro,1970Atkins,1970Urch,1972Hollas,1973Bishop,1977Vincent,1977Drago,1978Lowe,harris1978symmetry,1991Davidson,1993Diem,1993Lowe,1998Carter,2000Ohrn,2005Jacobs,2007Shirts"
literal "false"
\end_inset
if they have ever come across a list of all possible point groups for a
4-nucleus (tetra-nuclear or tetra-atomic) system, and in all cases we were
explicitly told that they were unaware of such a list, or we received no
response at all.
We have now determined all possible point groups for systems with four
nuclei and also for systems with
\emph on
any
\emph default
number of nuclei, and we have gone a step even further by determining all
possible point groups for structures with a given chemical formula.
\end_layout
\begin_layout Standard
\lang british
After searching through all of
\begin_inset CommandInset citation
LatexCommand citep
key "1955Wilson,1963Cotton,1966Hochstrasser,hall1969group,1969Ferraro,1970Atkins,1970Urch,1972Hollas,1973Bishop,1977Vincent,1977Drago,1978Lowe,harris1978symmetry,1991Davidson,1993Diem,1993Lowe,1998Carter,2000Ohrn,2005Jacobs,2007Shirts"
literal "false"
\end_inset
thoroughly, the closest work that we found to the results in this paper,
is presented in the table of
\begin_inset Quotes eld
\end_inset
shapes
\begin_inset Quotes erd
\end_inset
found in the 2008 version of the famous
\begin_inset Quotes eld
\end_inset
Tables for Group Theory
\begin_inset Quotes erd
\end_inset
document by P.
Atkins, M.
Child and C.
Phillips
\begin_inset CommandInset citation
LatexCommand citep
key "1970Atkins"
literal "false"
\end_inset
.
Nothing similar was found in any of
\begin_inset CommandInset citation
LatexCommand citep
key "1955Wilson,1963Cotton,1966Hochstrasser,hall1969group,1969Ferraro,1970Atkins,1970Urch,1972Hollas,1973Bishop,1977Vincent,1977Drago,1978Lowe,harris1978symmetry,1991Davidson,1993Diem,1993Lowe,1998Carter,2000Ohrn,2005Jacobs,2007Shirts"
literal "false"
\end_inset
, and the table of
\begin_inset Quotes eld
\end_inset
shapes
\begin_inset Quotes erd
\end_inset
in
\begin_inset CommandInset citation
LatexCommand citep
key "1970Atkins"
literal "false"
\end_inset
has several shortcomings which we will address and fix in Section
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:TableShapes"
plural "false"
caps "false"
noprefix "false"
\end_inset
, before we begin presenting the main results of our work in the following
sections.
\end_layout
\begin_layout Standard
\lang british
For investigating the novelty of our work, we believe that our thorough
examination through all of
\begin_inset CommandInset citation
LatexCommand citep
key "1955Wilson,1963Cotton,1966Hochstrasser,hall1969group,1969Ferraro,1970Atkins,1970Urch,1972Hollas,1973Bishop,1977Vincent,1977Drago,1978Lowe,harris1978symmetry,1991Davidson,1993Diem,1993Lowe,1998Carter,2000Ohrn,2005Jacobs,2007Shirts"
literal "false"
\end_inset
was a fair survey of the known literature on the topic.
The 2007 paper
\begin_inset CommandInset citation
LatexCommand citep
key "2007Shirts"
literal "false"
\end_inset
reported a thorough survey of known published
\begin_inset Note Note
status open
\begin_layout Plain Layout
\lang british
need better word for this
\end_layout
\end_inset
works that provided character tables for the
\begin_inset Formula $S_{8}$
\end_inset
and
\begin_inset Formula $D_{8h}$
\end_inset
point groups, and our list
\begin_inset CommandInset citation
LatexCommand citep
key "1955Wilson,1963Cotton,1966Hochstrasser,hall1969group,1969Ferraro,1970Atkins,1970Urch,1972Hollas,1973Bishop,1977Vincent,1977Drago,1978Lowe,harris1978symmetry,1991Davidson,1993Diem,1993Lowe,1998Carter,2000Ohrn,2005Jacobs,2007Shirts"
literal "false"
\end_inset
includes all of those works, in addition to
\begin_inset CommandInset citation
LatexCommand citep
key "2007Shirts"
literal "false"
\end_inset
itself, the papers that cited
\begin_inset CommandInset citation
LatexCommand citep
key "2007Shirts"
literal "false"
\end_inset
, and also thoroughly selected additional works.
\end_layout
\begin_layout Standard
Our paper focuses on
\emph on
axial
\emph default
point groups, which are those belonging to the series
\begin_inset Note Note
status open
\begin_layout Plain Layout
in what order shall we present these?
\end_layout
\end_inset
\begin_inset Formula $C_{n},C_{nv},C_{nh},S_{2n},D_{n},D_{nh}$
\end_inset
and
\begin_inset Formula $D_{nd}$
\end_inset
(including
\begin_inset Formula $C_{s}$
\end_inset
which is equivalent to
\begin_inset Formula $C_{1h}$
\end_inset
and
\begin_inset Formula $C_{1v}$
\end_inset
, and
\begin_inset Formula $C_{i}$
\end_inset
which is equivalent to
\begin_inset Formula $S_{2}$
\end_inset
).
For the
\emph on
polyhedral
\emph default
point groups
\lang british
\begin_inset Formula $T,T_{d},T_{h},O,O_{h},I,$
\end_inset
and
\begin_inset Formula $I_{h}$
\end_inset
,
\begin_inset Note Note
status open
\begin_layout Plain Layout
\lang british
what about
\begin_inset Formula $K_{h}$
\end_inset
?
\end_layout
\end_inset
we will present even more novel results in a forthcoming paper.
For example, in all of the literature that we have seen (including
\begin_inset CommandInset citation
LatexCommand citep
key "1955Wilson,1963Cotton,1966Hochstrasser,hall1969group,1969Ferraro,1970Atkins,1970Urch,1972Hollas,1973Bishop,1977Vincent,1977Drago,1978Lowe,harris1978symmetry,1991Davidson,1993Diem,1993Lowe,1998Carter,2000Ohrn,2005Jacobs,2007Shirts"
literal "false"
\end_inset
), the shape with the smallest number of vertices (or the system containing
the smallest number of nuclei) for which the point group is
\begin_inset Formula $O$
\end_inset
, is the snub cube which has 24 vertices (or nuclei), but within the last
month we have discovered that it is possible to construct an object with
only 20 vertices (or for the nuclei in a 20-nucleus system to be arranged)
such that the point group is
\begin_inset Formula $O$
\end_inset
\begin_inset CommandInset citation
LatexCommand citep
key "DattaniCunningham"
literal "false"
\end_inset
.
Likewise, in all of the literature that we have seen, the shape with the
smallest number of vertices for which the point group is
\begin_inset Formula $I$
\end_inset
, is the snub dodecahedron which has 60 vertices, but within the last month
we have discovered that it is possible to construct an object with only
50 vertices such that the point group is
\begin_inset Formula $I$
\end_inset
\begin_inset CommandInset citation
LatexCommand citep
key "DattaniCunningham"
literal "false"
\end_inset
.
\end_layout
\begin_layout Standard
\lang british
The next section of this paper (Section
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Motivation"
plural "false"
caps "false"
noprefix "false"
\end_inset
) will explain the context which made
\series bold
\emph on
us
\series default
\emph default
feel compelled to answer the questions that this paper addresses; for any
readers that wish to skip the motivation section, the rest of the paper
is self-contained and does not require knowledge of any of the material
presented in the next section.
The second paragraph of this introduction already described the contents
of Section
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:TableShapes"
plural "false"
caps "false"
noprefix "false"
\end_inset
.
The diagrams in Section
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:TableShapes"
plural "false"
caps "false"
noprefix "false"
\end_inset
will make it easier to follow Section
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:GeneralPrinciples"
plural "false"
caps "false"
noprefix "false"
\end_inset
, which contains the foundation for obtaining our main results which are
provided in Section
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:NumberOfAtomsForEachPointGroup"
plural "false"
caps "false"
noprefix "false"
\end_inset
and
\lang american
summarized
\lang british
in Section
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Summary"
plural "false"
caps "false"
noprefix "false"
\end_inset
.
Overall our results are
\lang american
outlined
\lang british
below, and all results appear to be novel or unusually difficult to find
in existing places.
\end_layout
\begin_layout Enumerate
\lang british
\begin_inset Argument 1
status open
\begin_layout Plain Layout
\lang british
leftmargin=0.35cm
\end_layout
\end_inset
In Section
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:TableShapes"
plural "false"
caps "false"
noprefix "false"
\end_inset
, we provide a table
\begin_inset Note Note
status open
\begin_layout Plain Layout
\lang british
Actually reference it
\end_layout
\end_inset
that presents visual examples of structures for all aforementioned axial
point groups with
\begin_inset Formula $2\le n\le6$
\end_inset
and
\begin_inset Formula $n=\infty,$
\end_inset
\end_layout
\begin_deeper
\begin_layout Enumerate
\lang british
\begin_inset Argument 1
status open
\begin_layout Plain Layout
\lang british
leftmargin=0.6cm
\end_layout
\end_inset
in a more clear way than in
\begin_inset CommandInset citation
LatexCommand citep
key "1970Atkins"
literal "false"
\end_inset
,
\end_layout
\begin_layout Enumerate
\lang british
with inconsistencies in
\begin_inset CommandInset citation
LatexCommand citep
key "1970Atkins"
literal "false"
\end_inset
resolved,
\end_layout
\begin_layout Enumerate
\lang british
with more of the
\begin_inset Formula $n=\infty$
\end_inset
shapes presented than in
\begin_inset CommandInset citation
LatexCommand citep
key "1970Atkins"
literal "false"
\end_inset
,
\end_layout
\begin_layout Enumerate
\lang british
with an extra column showing additional structures for the
\begin_inset Formula $n=2$
\end_inset
cases (in some cases these are more realistic structures than in
\begin_inset CommandInset citation
LatexCommand citep
key "1970Atkins"
literal "false"
\end_inset
),
\end_layout
\begin_layout Enumerate
\lang british
with structures requiring fewer nuclei than in
\begin_inset CommandInset citation
LatexCommand citep
key "1970Atkins"
literal "false"
\end_inset
for the
\begin_inset Formula $C_{n}$
\end_inset
and
\begin_inset Formula $D_{n}$
\end_inset
series, and
\end_layout
\begin_layout Enumerate
\lang british
with a more logical order, especially in terms of pedagogical considerations,
than in
\begin_inset CommandInset citation
LatexCommand citep
key "1970Atkins"
literal "false"
\end_inset
.
\end_layout
\end_deeper
\begin_layout Enumerate
\lang british
In Section
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:GeneralPrinciples"
plural "false"
caps "false"
noprefix "false"
\end_inset
, we describe some properties of various point groups and point group elements.
We have found that formulating
\begin_inset Note Note
status open
\begin_layout Plain Layout
\lang british
better word?
\end_layout
\end_inset
these properties was very helpful for obtaining or explaining new results,
and very helpful for pedagogical purposes.
\end_layout
\begin_layout Enumerate
\lang british
In Section
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:NumberOfAtomsForEachPointGroup"
plural "false"
caps "false"
noprefix "false"
\end_inset
, for each axial point group, we determine:
\end_layout
\begin_deeper
\begin_layout Enumerate
\lang british
\begin_inset Argument 1
status open
\begin_layout Plain Layout
\lang british
leftmargin=0.6cm
\end_layout