1.1 INTRODUCTION
We will already be familiar with the concept of
symmetry in an everyday sense. If we say something is ‘symmetrical’, we usually
mean it has mirror symmetry, or ‘left-right’ symmetry, and would look the same
if viewed in a mirror. Symmetry is also very important in chemistry. Some
molecules are clearly ‘more symmetrical’ than others, but what consequences
does this have, if any? The aim of this course is to provide a systematic
treatment of symmetry in chemical systems within the mathematical framework
known as group theory. Once we have classified the symmetry of a molecule,
group theory provides a powerful set of tools that provide us with considerable
insight into many of its chemical and physical properties.
Symmetry (from Greek συμμετρία symmetria "agreement
in dimensions, due proportion, arrangement") in
everyday language refers to a sense of harmonious and beautiful
proportion and balance. In mathematics, "symmetry" has
a more precise definition, that an object is invariant to a
transformation, such as reflection but including other transforms too. Although
these two meanings of "symmetry" can sometimes be told apart, they
are related, so they are here discussed together.
Mathematical
symmetry may be observed with respect to the passage of time;
as a spatial relationship;
through geometric transformations such
as scaling, reflection, and rotation; through other kinds
of functional transformations; and as an aspect of abstract objects, theoretic models, language, music and
even knowledge itself.
This
article describes symmetry from four perspectives: in geometry,
the most familiar type of symmetry for many people; more generally, in mathematics as a whole; as it relates to science and nature;
and in the arts, covering architecture, art and music.
The
opposite of symmetry is asymmetry.
In
chemistry, a property manifested in the geometrical configuration of molecules
and affecting the physical and chemical properties of molecules in the isolated
state, in an external field, and in interactions with other atoms and
molecules.
Most
simple molecules possess such elements of spatial symmetry in the equilibrium
configuration as axes of symmetry and symmetry planes. Thus, a molecule of
ammonia (NH3) possesses the symmetry of a regular triangular pyramid, while a
molecule of methane (CH4) possesses the symmetry of a tetrahedron. In complex
molecules, symmetry of the equilibrium configuration of the molecule as a whole
is usually absent although to a great extent the symmetry of the individual
molecular fragments (local symmetry) is preserved. The most complete
description of symmetry of both equilibrium and non-equilibrium configurations
of molecules is obtained from the concept of dynamic symmetry groups, that is,
groups that include not only operations of spatial symmetry of the nuclear
configuration but also operations involving the transposition of identical
nuclei in various configurations. For example, the dynamic symmetry group for
the NH3 molecule also includes the operation of inversion of this molecule: the
transfer of the N atom from one side of the plane formed by the H atoms to the
other.
The
symmetry of the equilibrium configuration of the nuclei in a molecule
determines the symmetry of the wave functions for various states of the
molecule. This relationship permits a classification of states according to
symmetry types. The transition between two states is related to the absorption
or emission of light; depending on the symmetry types of the states, the
transition either will be seen in the molecular spectrum or else will be
forbidden, in which case the line or band corresponding to this transition will
be absent in the spectrum. The symmetry types of states between which
transitions are possible affect both the intensity and the polarization of
spectral lines and bands. For example, in homo-nuclear diatomic molecules,
transitions between electronic states of identical parity, the electronic wave
functions of which behave identically upon inversion, are forbidden and do not
appear in the spectra. Also, in molecules of benzene and analogous compounds, transitions
between non degenerate electronic states of the same symmetry type are also
forbidden. The rules for selection according to symmetry are complemented for
transitions between different states by selection rules related to the spin of
these states.
In
molecules with paramagnetic centres, the symmetry of the environment of these
centres leads to a given type of anisotropy of the g-factor (Landé splitting
factor); this anisotropy affects the structure of the electron paramagnetic
resonance spectrum. In molecules with nuclei possessing nonzero spin, the
symmetry of the individual local fragments leads to a distinct type of
splitting of the energy states with different projections of nuclear spin; this
splitting affects the structure of nuclear magnetic resonance spectrum.
In
the approximate methods of quantum chemistry, which make use of the concept of
molecular orbitals, classification according to symmetry is possible not only
for the wave function of the molecule as a whole but also for the individual
orbitals. If the equilibrium configuration of a molecule has a symmetry plane
containing nuclei, all the orbitals of the molecule will fall into one of two
classes, being either symmetrical (σ)or anti-symmetrical (π) relative to the
operation of reflection in this plane. Molecules in which m orbitals are the
highest (with respect to energy), occupied orbitals form specific classes of
unsaturated and conjugated compounds with characteristic properties. Knowledge
of the local symmetry of the individual fragments of a molecule and of the
molecular orbitals localized on these fragments permits an evaluation of which
fragments will more readily undergo excitation and will be more strongly
altered in the course of chemical transformations, for example in photochemical reactions.
Symmetry
concepts have great importance in the theoretical analysis of the structure,
properties, and behaviour in various reactions of complex compounds. Crystal
field theory and ligand field theory postulate a mutual arrangement of the
occupied and vacant orbitals of a complex compound on the basis of data on the
compound’s symmetry. The theories also postulate the nature and degree of the
splitting of energy levels upon a change in the symmetry of the ligand field.
Knowledge of only the symmetry of a complex very often permits a qualitative
evaluation of the complex’s properties.
In
1965, R. Woodward and R. Hoffmann proposed the principle of conservation of
orbital symmetry in chemical reactions. The principle has been confirmed by
experiment and has had a great impact on the development of the branch of
organic chemistry dealing with the preparation of substances. Known as the
Woodward-Hoffmann rule, it states that the individual steps of a chemical
reaction proceed with the conservation of the symmetry of the molecular
orbitals, or orbital symmetry. The greater the violation of orbital symmetry in
a particular step, the more difficult it is for the reaction to proceed.
Consideration
of the symmetry of molecules is important in selecting the materials used in
producing chemical lasers and molecular rectifiers, building models of organic
superconductors, and analysing carcinogenic and pharmacologically active
substances.
2.1 IMPORTANCE OF SYMMETRY OPERATIONS IN
CHEMICAL PROPERTIES AND CHEMICAL REACTIONS
Symmetry
is important to chemistry because it undergirds essentially all specific
interactions between molecules in nature (i.e., via the interaction of natural
and human-made chiral molecules with inherently chiral biological systems). The
control of the symmetry of molecules produced in modern chemical synthesis
contributes to the ability of scientists to offer therapeutic interventions
with minimal side effects. A rigorous understanding of symmetry explains
fundamental observations in quantum chemistry, and in the applied areas of
spectroscopy and crystallography. The theory and application of symmetry to
these areas of physical science draws heavily on the mathematical area of group
theory. This is the reason why symmetry operations and symmetry elements in a
molecule are important to the inorganic chemistry.
2.1.1 Constructing Molecular and Hybrid Orbitals
In
chemistry, a molecular orbital (or MO) is a mathematical function describing
the wave-like behaviour of an electron in a molecule. This function can be used
to calculate chemical and physical properties such as the probability of
finding an electron in any specific region. The term orbital was introduced by
Robert S. Mulliken in 1932 as an abbreviation for one-electron orbital wave
function.[1] At an elementary level it is used to describe the region of space
in which the function has a significant amplitude. Molecular orbitals are
usually constructed by combining atomic orbitals or hybrid orbitals from each
atom of the molecule, or other molecular orbitals from groups of atoms. They
can be quantitatively calculated using the Hartree–Fock or self-consistent
field (SCF) methods.
A
molecular orbital (MO) can be used to represent the regions in a molecule where
an electron occupying that orbital is likely to be found. Molecular orbitals
are obtained from the combination of atomic orbitals, which predict the location
of an electron in an atom. A molecular orbital can specify the electron
configuration of a molecule: the spatial distribution and energy of one (or one
pair of) electron(s). Most commonly an MO is represented as a linear
combination of atomic orbitals (the LCAO-MO method), especially in qualitative
or very approximate usage. They are invaluable in providing a simple model of
bonding in molecules, understood through molecular orbital theory. Most
present-day methods in computational chemistry begin by calculating the MOs of
the system. A molecular orbital describes the behaviour of one electron in the
electric field generated by the nuclei and some average distribution of the
other electrons. In the case of two electrons occupying the same orbital, the
Pauli principle demands that they have opposite spin. Necessarily this is an
approximation, and highly accurate descriptions of the molecular electronic
wave function do not have orbitals.
Molecular
orbitals arise from allowed interactions between atomic orbitals, which are
allowed if the symmetries (determined from group theory) of the atomic orbitals
are compatible with each other. Efficiency of atomic orbital interactions is
determined from the overlap (a measure of how well two orbitals constructively
interact with one another) between two atomic orbitals, which is significant if
the atomic orbitals are close in energy. Finally, the number of molecular
orbitals that form must equal the number of atomic orbitals in the atoms being
combined to form the molecule.
Molecular
orbitals were first introduced by Friedrich Hund and Robert S. Mulliken in 1927
and 1928. The linear combination of atomic orbitals or "LCAO"
approximation for molecular orbitals was introduced in 1929 by Sir John
Lennard-Jones.[8] His ground-breaking paper showed how to derive the electronic
structure of the fluorine and oxygen molecules from quantum principles. This
qualitative approach to molecular orbital theory is part of the start of modern
quantum chemistry. Linear combinations of atomic orbitals (LCAO) can be used to
estimate the molecular orbitals that are formed upon bonding between the
molecule's constituent atoms. Similar to an atomic orbital, a Schrödinger
equation, which describes the behaviour of an electron, can be constructed for
a molecular orbital as well. Linear combinations of atomic orbitals, or the
sums and differences of the atomic wave functions, provide approximate
solutions to the Hartree–Fock equations which correspond to the
independent-particle approximation of the molecular Schrödinger equation. As
the two atoms become closer together, their atomic orbitals overlap to produce
areas of high electron density, and, as a consequence, molecular orbitals are
formed between the two atoms. The atoms are held together by the electrostatic
attraction between the positively charged nuclei and the negatively charged
electrons occupying bonding molecular orbitals.
A
MO with σ symmetry results from the interaction of either two atomic s-orbitals
or two atomic pz-orbitals. An MO will have σ-symmetry if the orbital is
symmetrical with respect to the axis joining the two nuclear centres, the
inter-nuclear axis. This means that rotation of the MO about the inter-nuclear
axis does not result in a phase change. A σ* orbital, sigma anti-bonding
orbital, also maintains the same phase when rotated about the inter-nuclear
axis. The σ* orbital has a nodal plane that is between the nuclei and
perpendicular to the inter-nuclear axis.
2.1.2 Interpreting Spectroscopic
2.1.2.1 Vibrational Spectroscopy
Infrared
(IR) and Raman spectroscopies are branches of vibrational spectroscopy and the
former technique is much the more widely available of the two in student
teaching laboratories. Vibrational
spectroscopy is concerned with the observation of the degrees of vibrational
freedom, the number of which can be
determined as follows. the motion of the molecule containing n atoms can conveniently be described in
terms of three Cartesian axes, the molecule has 3n degrees of freedom which
together describe the translational, vibrational and rotational motions of the
molecule.
2.1.2.2 Raman Spectroscopy
Chandrasekhara
V. Raman was awarded the 1930 Nobel Prize in Physics ‘for his work on the
scattering of light and for the discovery of the effect named after him’. When
radiation usually from the laser of a particular frequency, v0, falls on a
vibrating molecule, most of the radiation is scattered without change in
frequency. This is called Rayleigh scattering. One of the advantages of Raman
spectroscopy is that is extends to lower wave numbers than routine laboratory
IR spectroscopy, thereby permitting the observation of, for example,
metal-ligand vibrational modes. A disadvantage of the Raman effect is its
insentivity since only a tiny percentage of the scattered radiation undergoes
Raman scattering. one way of overcoming this is to use a Fourier transform (FT
technique). A second way, suitable only for coloured compound, is used to
resonance Raman spectroscopy. This technique realise on using laser excitation
wavelength that co-inside with wavelengths of absorbtion in the electronic
spectrum of a compound. This leads to resonance enhancement and increase in the
intensity of lines in the Raman spectrum. Resonance Raman spectroscopy is now
used extensively for the investigation of coloured d-block metal complexes and
for probing the active metals side in metalloproteins.
2.1.2.3 Predict Whether a Given Molecule is Chiral
One
example of symmetry in chemistry that you will already have come across is
found in the isomeric pairs of molecules called enantiomers. Enantiomers are
non-superimposable mirror images of each other, and one consequence of this
symmetrical relationship is that they rotate the plane of polarized light
passing through them in opposite directions. Such molecules are said to be
chiral, meaning that they cannot be superimposed on their mirror image. Formally,
the symmetry element that precludes a molecule from being chiral is a
rotation-reflection axis. Chiral molecule can rotate the plane of
plane-polarized light. This property is known as optical activity and the two
mirror images are known as optical isomers or enantiomers.
The
importance of chirality is clearly seen in, for example, dramatic differences
in the activities of different enantiomers of chiral drugs. A helical chain is
easy to recognize, but it is not always search a fasile task to identify a chiral
compound by attempting to convince oneself that it is, or is not,
non-superposable on its mirror image. symmetry consideration come to our aid; a
chiral molecular species must lack an improper (Sn) axis of
symmetry.
Another
commonly used criterion of an inversion centre, i, and plane of symmetry, σ. However, both of this properties are
compatible with the criterion given above, since we can rewrite the symmetry
operation i and σ in terms of the
improper rotation S2 and S1 respectively.
3.1 SUMMARY
Symmetry
is a property of molecules having more than one atom of the same kind, with
equal bond lengths and/or bond angles. As like the high symmetry of the SF6
molecule arises from the six equal S-F bonds disposed at angles of 90°
to each other. In order to build the notion more precise we use the idea of a
symmetry operation. For example rotating SF6 by 90° about an appropriate axis,
it appears indistinguishable after the rotation. The axis concerned is known as
the symmetry element. Rotations which do not leave the molecule looking the
similar are not symmetry operations.
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