Symmetry and Molecular Chirality: Lord Kelvins Legacy Laurence Barron

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Symmetry and Molecular Chirality Lord Kelvins
LegacyLaurence Barron
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Chirality
Chirality, meaning right- or left-handedness,
pervades much of modern science from the physics
of elementary particles, through organic
stereochemistry, to the structure and behaviour
of the molecules of life, with much else besides
(nonlinear optics nanotechnology electrical
engineering pharmaceuticals etc.). Life is
based on homochiral chemistry! Hence the
importance of chirality in the origin of life and
astrobiology.
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Lord Kelvins Definition of Chirality
I call any geometrical figure or group of points
chiral, and say that it has chirality if its
image in a plane mirror, ideally realized, cannot
be brought into coincidence with itself. (Lord
Kelvin, Baltimore Lectures, 1904)
Mirror-image chiral molecules (enantiomers) show
optical rotation of equal magnitude but opposite
sign. (Louis Pasteur, 1848)
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Natural and Magnetic Optical Activity
Isotropic collections of chiral molecules (e.g. a
sugar solution) show natural optical activity
phenomena such as optical rotation. A magnetic
field will induce magnetic optical activity in
achiral samples. E.g. a static magnetic field
parallel to the incident light beam induces
optical rotation in collections of achiral
molecules (the Faraday effect).
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Proteins
Proteins consist of polypeptide chains made from
combinations of 20 different amino acids, all
exclusively the L-enantiomers. Each different
protein has a unique polypeptide sequence of
several hundred amino acids which folds into a
unique three-dimensional (chiral) structure in
the native state.
Right-handed a-helix from L-amino acids
(secondary structure)
Tertiary structure (fold) of a native protein
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Nucleic Acids
Built from chains of deoxyribonucleosides (DNA)
or ribonucleosides (RNA), connected by
phosphodiester links, with four different bases.
Only D-ribose sugars occur.
Right-handed B-type DNA double helix from
D-ribose sugars
D-ribose
Deoxyribonucleic acid (DNA)
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Homochirality is a Sine Qua Non for Lifes
Chemistry!

• Molecules of sufficient complexity to support
  life on Earth inevitably exist in two
  mirror-image chiral forms. From the lock and
  key aspect of biomolecular recognition processes
  such as enzyme catalysis, where there is
  chirality there must be homochirality for an
  efficient biochemistry.
• Hence homochirality appears to be an inevitable
  characteristic of the chemistry of any
  molecule-based life.

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Carbon, Water and Life

• No element other than carbon forms such a huge
  variety of compounds, many of them chiral. No
  other element has such a propensity for
  concatenation (formation of long chains or
  rings), especially important for constructing the
  backbones of biopolymers.
• Liquid water is the absolutely essential medium
  for life on Earth. It also acts as a lubricant
  of key biochemical molecular processes. No other
  solvent can be envisaged having the necessary
  physico-chemical properties- they are unique to
  liquid water.
• Hence life on other worlds (if it exists) will
  probably be based on a homochiral organic
  chemistry in an aqueous environment, even if it
  originated independently from life on earth.

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Which Came First Homochirality in the Prebiotic
Monomers or in the Earliest Prebiotic Polymers

• Homochiral nucleic acid polymers, for example, do
  not form efficiently in a racemic solution of
  monomers due to enantiomeric cross inhibition
  (Orgel). But since enantiomeric cross-inhibition
  selectively inhibits formation of heterochiral
  polymers, then mutatis mutandis it will
  selectively amplify formation of homochiral
  polymers (Sandars).
• However, homochirality in the chiral monomers is
  not essential for generating homochiral synthetic
  polymers. A small chiral bias may suffice, as in
  sergeants-and-soldiers and majority rules effects
  in polyisocyanates (Green).
• Furthermore, a small chiral bias in the form of a
  few percent enantiomeric excess (ee) of one
  enantiomer can generate homochiral monomers in
  solid-liquid phase equilibria (Blackmond) and in
  sublimation (Cooks, Feringa) of racemic amino
  acids.
• Kinetically-induced amplification via
  autocatalytic reactions.
• Additional possibilities from crystal and surface
  chemistry (Lahav).

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Pasteurs Conjecture

• Small initial ees in chiral monomers can generate
  large ees in both chiral monomers and polymers.
  This small ee could be produced by some physical
  chiral influence.
• Pasteur (1884) conjectured that molecular
  chirality in the living world is the product of
  some universal chiral force or influence in
  nature. (At the time all substances found to be
  optically active in solution were natural
  products.)

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Early Attempts to Generalize the Chirality Concept
In looking for his universal chiral force, Louis
Pasteur (1884) attempted to extend the concept of
chirality (which he called dissymmetry) to other
aspects of the physical world, e.g.
Translation plus rotation
Magnetic field (since it induces optical
rotation, the Faraday effect)
B
Pierre Curie (1894) suggested that collinear
electric and magnetic fields are dissymmetric
(chiral)
E
-E
or
B
B
But only translation plus rotation constitutes
true chirality!
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Fundamental Symmetries

• Physical laws should be invariant under three
  fundamental symmetry operations
• Parity P inverts the positions of all the
  particles in a system through an arbitrary
  space-fixed origin (x,y,z -x,-y,-z).
• Time reversal T reverses the direction of motion
  of all the particles (t -t).
• Charge conjugation C interconverts particles and
  antiparticles.
• Physical quantities are classified accordingly
• Scalar and pseudoscalar quantities have no
  directional properties. The first are invariant
  under P, whereas the second change sign.
• Polar vector and axial vector quantities are
  associated with one spatial direction. The first
  change sign under P, whereas the second are
  invariant.
• Time-even and time-odd quantities are invariant
  under T, or change sign, respectively.

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Polar and Axial Vectors are Distinguished by
Behaviour under Parity
Parity P inverts the system through the origin of
arbitrary space-fixed axes.
Polar vectors change sign under P (e.g. position
r, velocity v, linear momentum p, electric dipole
m).
Axial vectors do not change sign under P (e.g.
angular momentum L, magnetic dipole m).
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Electric and Magnetic Fields have Opposite
Behaviour under Parity and Time Reversal

• Electric field E may be generated by two
  oppositely charged plates.
• P exchanges the position of the plates.
• T has no effect.
• Hence E is a polar time-even vector.
• Magnetic field B may be generated by a
  cylindrical current sheet.
• P has no effect.
• T reverses the direction of the current.
• Hence B is an axial time-odd vector.

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Symmetry and Optical Rotation
The magnetic rotation has neither right-handed
nor left-handed quality (that is to say, no
chirality). This was perfectly understood by
Faraday and made clear in his writings, yet even
to the present day we frequently find the chiral
rotation and the magnetic rotation of the plane
of polarised light classed together in a manner
against which Faradays original description of
his discovery contains ample warning. Lord
Kelvin (Baltimore Lectures, 1904)

• Chiral phenomena such as natural optical rotation
  are characterized by time-even pseudoscalar
  observables. The quantum states of the system
  must have mixed parity and definite reversality
  like Y(L) and Y(R).
• Magnetic optical rotation is not a chiral
  phenomenon (the observable is a time-odd axial
  vector). The quantum states must have definite
  parity and mixed reversality like Y(J,M) and
  Y(J,-M).

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The Mixed Parity States of a Chiral Molecule
The handed states are states of broken parity
that are interconverted by P. The Hamiltonian has
inversion symmetry but the handed states do not.
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Chiral Symmetry
The terms chiral symmetry and chiral symmetry
breaking, which are widely used to describe the
appearance of chirality out of achiral
precursors, are inappropriate because chirality
is not a symmetry at all in molecular science.
Rather, chirality is an attribute associated with
special types of reduced spatial symmetry that
enables an object to exist in two nonsuperposable
mirror-image forms. Mirror symmetry breaking is
more correct (Walba, Cintas). The term chiral
symmetry breaking is, however, entirely
appropriate in elementary particle physics, which
requires relativistic quantum field theory within
which chiral symmetry has a rigorous definition.
Chiral symmetry is an internal symmetry, rather
than a geometrical symmetry, of massless
particles, with mass associated with broken
chiral symmetry.
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True and False Chirality
Chirality in a stationary system such as a helix
or a chiral molecule is easy to recognize. But
when motion is an essential ingredient two types
of enantiomorphism must be distinguished

• True chirality (time-invariant
  enantiomorphism).
• Refers to systems that exist in two distinct
  enantiomeric states that are interconverted by
  parity P but not by time reversal T combined with
  any spatial rotation. Supports time-even
  pseudoscalar observables. A truly chiral
  influence can induce absolute enantioselection in
  all circumstances (lifts degeneracy of
  enantiomers).
• Breaks parity P but not time reversal T.
• False chirality (time-noninvariant
  enantiomorphism).
• Refers to systems that again exist in two
  distinct enantiomeric states, but now they are
  interconverted by T as well as P. Does not
  support time-even pseudoscalar observables. A
  falsely chiral influence cannot induce absolute
  enantioselection in a system at equilibrium (does
  not lift degeneracy of enantiomers).
• Breaks both P and T separately but is
  PT-invariant overall.
• L.D. Barron, J. Am. Chem. Soc. 108, 5539 (1986).

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Chirality of Spinning Particles
A stationary spinning electron is not a chiral
object
But it becomes truly chiral if translating with
its spin projection parallel or antiparallel to
the propagation direction
i.e. time-invariant enantiomorphism A similar
diagram describes a circularly polarized photon,
which is always truly chiral since photons always
move at the velocity of light.
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Collinear Electric and Magnetic Fields
Curie (1894) pointed out that collinear electric
and magnetic fields generate spatial dissymmetry
(enantiomorphism). Since E is a polar vector and
B is an axial vector, parallel and antiparallel
arrangements are interconverted by P
E
-E
P
B
B
But E is time-even and B is time-odd, so they are
also interconverted by T
E
E
T
B
-B
The apparent chirality is therefore false. i.e.
time-noninvariant enantiomorphism. Breaks P and T
separately but is PT-invariant overall.
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Magnetochiral Phenomena
A magnetic field B collinear with the propagation
vector k of an unpolarized light beam constitutes
a truly chiral system. Since k is a polar vector
and B is an axial vector, parallel and
antiparallel arrangements are interconverted by P
k
-k
P
B
B
But k is and B are both time-odd, they are not
interconverted by T
k
-k
T
B
-B
i.e. time-invariant enantiomorphism. Breaks P
but not T.
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Magnetochiral Dichroism
Chiral molecules absorb unpolarized light
differently in a static magnetic field parallel
and antiparallel to the light beam (Rikken and
Raupach 1997).
The unpolarized light beam parallel or
antiparallel to the magnetic field constitutes a
truly chiral influence. Rikken and Raupach (2000)
exploited this to induce absolute
enantioselection. See L. D. Barron, Nature, 405,
895 (2000).
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Enantioselective Magnetochiral Photochemistry
Enantioselection has been induced using a
magnetic field collinear with the propagation
direction of an unpolarized light beam
(magnetochiral influence- truly chiral). Neither
the magnetic field nor the unpolarized light beam
alone are enantioselective.
Cr(ox)33-
With the field parallel to the light beam a small
excess of one enantiomer is induced with the
field antiparallel an equal excess of the other
enantiomer is induced. (G.L.J.A. Rikken and E.
Raupach, 2000. Nature 405, 932.)
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Parity Violation (Cosmic Chirality)
In the weak interactions (responsible for
radioactive b-decay among other things), nature
makes an absolute distinction between right- and
left-handed spin-polarized particles. Parity P is
therefore not a true symmetry of the world
An extreme example is that only left-handed
neutrinos and right-handed antineutrinos exist.
So is parity P plus charge conjugation C
(particle-antiparticle exchange) the true
symmetry?
antineutrino
neutrino
(In fact CP itself can sometimes be violated,
leaving CPT as the true symmetry.)
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The Parity-Violating Weak Neutral Current
Interaction
The unified theory of the weak and
electromagnetic interactions reveals that parity
violation infiltrates to a tiny extent into all
electromagnetic phenomena. The weak neutral
current interaction generates the following
parity-violating electron-nucleus contact
interaction in the Hamiltonian
se Pauli spin operator of the electron
(time-odd axial vector) pe linear momentum
of the electron (time-odd polar vector) rN
nuclear density function QW effective weak
charge
se..pe is a time-even pseudoscalar. It is the
quintessential truly chiral influence in atomic
and molecular physics! E.g. free atomic vapours
show tiny parity-violating optical rotations. An
atom and its anti-atom (generated by CP ) show
equal and opposite optical rotations.
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Parity Violation and Chiral Molecules
The weak neutral current interaction generates a
tiny parity-violating energy difference (10-15
kJ mol-1) between the mirror-image enantiomers
of a chiral molecule, which are therefore not
strict enantiomers
The strict enantiomer (identical energy) of a
chiral molecule is the mirror-image molecule
composed of antiparticles. (L.D. Barron, 1981.
Chem. Phys. Lett. 79, 392). Holds even if CP
is violated, provided CPT is conserved. (L.D.
Barron, 1994. Chem. Phys. Lett. 221, 311).
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Cosmic Chirality


Having conjectured that molecular chirality in
the living world is the product of universal
chiral forces in nature, Pasteur would have been
entranced by the discovery of parity violation in
the fabric of the universe, and the way in which
it lifts the degeneracy of mirror-image chiral
molecules. Whether or not parity violation has
anything to do with the origin of biomolecular
homochirality and the origin of life is the
subject of much debate!
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CP Violation
Decay rate asymmetry of the long-lived neutral
K-meson     The two sets of decay products are
interconverted by CP   Hence CP is violated.
This automatically implies T violation from the
CPT theorem (the Hamiltonian is invariant to CPT
even if it is not invariant to one or more of
these operations).
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Direct T Violation
Direct T violation was observed much later in the
form of slightly different rates for the
particle-to-antiparticle and reverse
processes This implies that T violation
corresponds to a breakdown in microscopic
reversibility. Since a particle and its
antiparticle have the same rest mass if CPT is
conserved, only the kinetics but not the
thermodynamics are affected in CP- or T-violating
processes.
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Absolute Enantioselection
Consider a unimolecular process in which an
achiral molecule R generates a chiral molecule M
or its enantiomer M
In the absence of a chiral influence M and M
have the same energy, so no ee can exist if the
reaction reaches thermodynamic equilibrium.
A truly chiral influence lifts the degeneracy of
M and M so an ee of one or other can exist at
equilibrium.
A falsely chiral influence does not lift the
degeneracy of M and M so an ee cannot exist at
equilibrium.
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Microscopic Reversibility
Microscopic reversibility follows from the
behaviour of the quantum-mechanical transition
amplitude under time reversal. For a transition
from some initial linear momentum state p (of one
or more interacting particles) to some final
state p we have
This implies the same P.E. barriers for the
forward and reverse processes
The principle of detailed balancing follows from
a quantum-statistical average over all the
colliding particles at thermal equilibrium.
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Enantiomeric Microscopic Reversibility
For molecules in a falsely chiral influence like
collinear E and B, time reversal is no longer a
symmetry operation because it generates the
enantiomorphous influence. Hence conventional
microscopic reversibility and detailed balancing
no longer hold. However, application of P as well
as T restores the influence to its original state
(apart from a rotation through p).
E
-E
E
B
-B
-B
The star denotes the P-enantiomer. The first
equality accords with Onsagers prescription
(1931) that microscopic reversibility is
recovered if the magnetic field direction is
reversed along with the particle velocities. But
in a given experiment the relative field
directions remain fixed, which implies a
breakdown in conventional microscopic
reversibility. Symmetry is recovered in the
corresponding time-reversed enantiomeric process!
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Hence for reacting molecules in a falsely chiral
influence we only have enantiomeric microscopic
reversibility based on
M R
R M
This implies the same P.E. barriers for the
forward and reverse enantiomeric processes, but
different barriers for the forward and reverse
processes involving the same enantiomer
For achiral molecules M M and the two barriers
coalesce, becoming identical in the forward and
reverse directions.
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An Example Conrotatory (Thermal) Ring Closure of
a Butadiene
Illustrates the requirement of a net circulation
of charge in a plane perpendicular to B, with E
serving to align the molecule in the magnetic
field. Equal and opposite velocity-dependent
contributions to the P.E. surface are generated
for the two senses of circulation. (L.D. Barron,
1987. Chem. Phys. Lett. 135, 1.)
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Absolute Enantioselection Induced by a Falsely
Chiral Influence Analogy with CP Violation
in the presence of collinear E and B.
CP violation is revealed by the observation
. This decay rate asymmetry can therefore
be conceptualized as arising from a breakdown in
microscopic reversibility due to a
time-noninvariant CP-enantiomorphous influence in
the forces of nature. (The CPT theorem guarantees
that the two distinct CP-enantiomorphous
influences are interconverted by T).
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True Thermodynamic Equilibrium
R R
Enantiomeric detailed balancing allows
and in a falsely chiral
influence. But this generates an ee which
conflicts with the requirement that M M
at equilibrium! This conflict between the
kinetic and thermodynamic requirements can be
resolved by including other racemization
pathways. Then at true thermodynamic equilibrium
(i.e. when all the possible infinite racemization
pathways have equilibrated), M M obtains
because the different ees associated with each
separate pathway sum to zero. Proof follows from
the unitarity of the scattering matrix (the
probabilities of all possible transitions to and
from a given state must sum to unity). It is in
fact unitarity, not microscopic reversibility,
that lies behind the validity of Boltzmanns
H-theorem and hence the second law.
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CP Violation, False Chirality and Catalysis
The force responsible for CP violation lacks CP
and T invariance separately but is CPT invariant
overall. This is analogous to a falsely chiral
influence such as collinear E and B, which is
characterized by lack of P and T invariance
separately but by PT invariance overall. The
force responsible for CP violation is the
quintessential falsely chiral influence in
particle physics!
A falsely chiral influence acts as a chiral
catalyst since it modifies P.E. barriers to
change relative rates of formation of
enantiomeric products without affecting their
relative energies and hence the equilibrium
thermodynamics.
From the CPT theorem, the rest mass of a particle
and its antiparticle are identical even if CP is
violated. CP violation is therefore analogous to
chemical catalysis since can affect rates of
particle-antiparticle processes (via a breakdown
in microscopic reversibility) without affecting
the thermodynamics!
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Summary Lord Kelvins (and Louis Pasteurs!)
Legacy

• We live in a chiral world, from the very fabric
  of the universe to the nuts and bolts of life.
• Everything in the universe is intertwined and
  basically inseparable.

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