Transcript Advanced Chemical Physics 1

Advanced Chemical Physics
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Advanced Physical Chemistry
Spectroscopy
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Electronic spectroscopy (basics in quantum mechanics)
Vibrational spectroscopy (IR+Raman)
Time resolved spectroscopy
Surface spectroscopy
Single molecule spectroscopy
Photoelectrons spectroscopy
Advanced Topics in Thermodynamics and Kinetics
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Liquid-solid interfaces (wetting, contact angle)
Molecules at Interfaces (Langmuir films, self-assembled layers)
Catalysis (Chemisorption, kinetics, mechanisms)
Structure and dynamics in liquids
Books: 1. Modern Spectroscopy, J. M. Hollas, John Wiley&Sons
2. Molecular Vibrations, Wilson, Decious and Cross, Dover
Publications Inc.
3. Physical Chemistry of Surfaces, A.W. Adamson and Cast, WileyInterscience Publication.
4. “An Introduction to the Liquid State" by P.A. Egelstaff, Oxford
University Press.
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Electronic Spectroscopy
• Quantum mechanics- Born-Oppenheimer approximation
• Molecular symmetry
• Electromagnetic radiation and its interaction with atoms
and molecules
• Coupling of angular momenta
• Classification of electronic states and selection rules.
• Vibronic spectra, Franck-Condon principle and selection
rules
• Non Born-Oppenheimer effects, radiationless transitions.
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Quantum mechanicsBorn-Oppenheimer approximation
In 1924 Louis de Broglie recognized the similarity that exists between Fermat’s principle of least time, which
governed the propagation of light, and Maupertuis’s principle of least action, which governed the propagation
of particles. He proposed that with any moving body there is associated a wave and that the momentum
of the particle and the wavelength are related by: p=h/.
It can be shown that as a result of this relation one obtains also the Heisenberg uncertainty principle:
p  x ≥h.
Hence in order that an electron will reside in a radius around a nuclei a standing wave must exist in which
n  r
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In quantum mechanics we deal with the solution of the Schrödinger Equation,
which is an equation for the spatial and temporal behavior of the de Broglie
waves:
The Schrödinger Equation is given by:
H  i t 



H    2 2m  2 x 2   2 y2   2 z 2  V
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The equation can be separated into two parts (and making it one dimensional):
 d2

 ( x )  V( x ) ( x )  E ( x )
2
2m dx
i
d
 ( t )  E ( t )
dt
Hence:
x, t   C ( x)e

iE t

The time independent equation has the form of a standing wave. The time dependent
part results in a phase, which does not effect the probability which is given by
(x,t) *(x,t), when *(x,t) is the complex conjugate of (x,t).
Most of the spectroscopy requires only to understand the standing wave
approximation.
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In the case of molecules, the Hamiltonian is very complex because it contains the kinetic energies and
interactions between many particles- all the nuclei and all the electrons.
For the hydrogen atom, and for hydrogen-like ions with a single electron in the field of a nucleus with
charge +Ze, the Hamiltonian is given by
 2
Ze 2
H
 
2
4 0 r
For polyelectronic atom the Hamiltonian becomes

Ze 2
e2
2
H
 i   4 r   4 r
2 i
0 i i j
0 ij
i
where the summation is over all electrons i. Because of the last term, the Hamiltonian cannot be
broken down into a sum of contributions from each electron and the Schrödinger equation can no
longer be solved exactly. To overcome this one uses the Hartree-Fock approximation, not to be
discussed here.
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In the case of polyatomic molecules, in addition to the electron-nuclei and electronelectron terms in the potential energy we have also the nucleus-nucleus repulsion in
the kinetic energy we have in addition to the electronic kinetic energy also the nuclei
kinetic energy. Hence,
H=Te+Tn+Ven+Vee+Vnn
In the Born Oppenheimer approximation one assumes that since the nuclei are much
heavier than the electrons, the electrons adjust instantly to any change in the nuclei
configuration (a “classical” way of thinking) and therefore one can solve the equation
for the electrons at each fixed configuration of the nuclei (can you see why this is
“wrong” in quantum mechanics?). Hence the nuclei configuration is serving as
parameters in the electronic equation.
Now the hamiltonian has the form of:
H= He+Hn
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Therefore it is possible now to solve the equation separately for the electrons and the
nuclei so that:
H e e(q, Q 0 )  E e (Q) e(q, Q 0 )
where He=Te+Ven+Vee and
Hn=Tn+Vnn+Ee when Ee contribute to the potential
energy of the nuclei due to the solution of the equation of the electrons.
So now the equation for the nuclei is given by:
H n n(Q)  E n n(Q)
This means that:
and
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 (q, Q)   e (q, Q) n (Q)
E=Ee+En
In order to solve the Schr ödinger equation the following operation has to be performed:
i Hi  i Ei i  Ei i i
now we ontegrate from both sides and obtain :
 i Hidt  E  i idt  Ei
In order that the integral will be no zero it is required that the expression i Hi will be
symmetric in t, hence I must be either symmetric or antisymmetric with regards to the
symmetry of H.
Hence if we know the symmetry of H we know something about the property of the
wavefunction.
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Molecular Symmetry
Elements of Symmetry-
n-fold axis of symmetry Cn
Plane of symmetry, sv; sh; sd (d=dihedral)
Center of inversion, i
n-fold rotation-reflection axis of symmetry, Sn sh x Cn=Sn
The identical elements, I
Conditions for chirality- A molecule is chiral if it does not have
any Sn symmetry element with any value of n.
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Point groups- All elements of symmetry which any molecule may have constitute a point
group.
Cn point group contains Cn axis of symmetry and Cn2, Cn3,.. Cnn-1
Cnv point group contains a Cn axis and n s planes of symmetry all of which contain
the Cn axis.
Dn point group contains a Cn axis and n C2 axes.
Cnh contains Cn axis and a sh plane perpendicular to Cn
Dnd point group contains a Cn axis, an S2n axis, n C2 axes perpendicular to
Cn and nsd planes (ethan in staggered configuration belongs to D3d).
Dnh point group contains a Cn axis, nC2 axes, a sh plane and n other s
planes. (benzen belong to D6h point group).
Td point group contains four C3 axes, three C2 axes, and six sd planes. (tetrahedral).
Character table
Character table of a point group summarized the way a state is transformed under a certain
symmetry operator belonging to this group.
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Electromagnetic radiation and its interaction with atoms
and molecules
Electromagnetic radiation, means that energy is moving in the form of both electric and
magnetic energies. The perturbed the irradiated region by oscillating fileds, where the
electric (E) and magnetic (H) fields are perpendicular to each other. For a radiation
travelling in the direction x the E and H vectors are in in the direction of y and z
respectively.
E y  A sin( 2t  kx)
H z  A sin( 2t  kx)
where A is the amplitude. Because k is the same for each component, they are in-phase.
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The plane of polarization is conventionally taken to be the
plane containing the direction of E and that of the propagation,
this is the plane xy.
This is because the interaction of radiation with matter is much
stronger through E than through H.
En
n
E
Em
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m
The rate of change of population Nn of state n due to induced absorption is given by

dN n
 N m Bmn  ( )
dt
Where Bnm is a the Einstein coefficient and the spectral radiation density is given by:

3
8hc

 ( ) 

exp( hc kT )  1
Similarly, induced emission changes the population of Nn by

dN n
  N m Bnm  ( )
dt
where Bnm=Bmn.
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For spontaneous emission
dN n
  N n Anm
dt
In the presence of radiation that has an energy of ~ wavenumbers, all processes occurring at
once. If the radiation is on, the population reaches equilibrium and
dN n
 N m  N n B nm  (~)  N n A nm  0
dt
At equilibrium the populations in the two states are related through the Boltzmann distribution
Nn gn
 E 

exp 

Nm gm
 kT 
From the above equations one obtains the ration between Anm and Bnm
Anm  8 hc~3Bnm
This equation illustrates that the spontaneous emission increases rapidly, relative to the induced
emission, as the excitation energy increases.
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Electric field can interact with atoms or molecules through the dipole operation (first
order).
The dipole is defined as k   qi k i
when qi is the charge of the i-th particle, ki is
i
the coordinate of the i-th particle and k=x,y,z.
The dipole transition moment has the form of
*
R nm
k   n  k m dk
and the transition probability is given by:
R
nm 2
 2
  R nm
k
It is related to Bnm by:
k
B nm 
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8 3
4 0 3h
2
R
nm 2
In absorption experiment I  I 0 exp  s ( ) Nl  when s is the cross section for the process, N
are the density of molecules and l is the length of the measuring system. The absorbance is
defined as:
I 
A  log10  0    ( )c
I 
Often one uses the term max however the physical meaningful value is the integrated area under
the peak assuming that Nn<<Nm (no stimulated emission).
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The relation between Bnm and the molar absorptivity is:
~2
N A h~nm B nm
~
~
.
~  ( ) d 
ln
10

1
NA is the Avogadro number.
If the absorption is due to electronic transition, sometimes one uses oscillator strength as a
measure for the strength of the transition
f nm 
4 0 m e c ln 10
2
NAe
2
~2
~  ( ) d
~
~
1
The quantity fnm is dimensionless and is the ratio of the strength of the transition to that of an
electric dipole transition between two states of an electron oscillating in three dimensions in a
simple harmonic way. The maximum value of fnm is usually one.
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Short Summary
The “dream” of any quantum chemist is to present
the hamiltonian in the form of:
H=h1+h2+ h3+….
When hi is hydrogen atom like hamiltonian.
The solution is given than as:
  1 2 3....
E=E1+E2+E3+…
Alternatively one tries to write the hamiltonian as
sum of hamiltonians with the terms that couple
them as “off-diagonal” terms.
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Short Summary-continue
The transition dipole moment
R mn   n*  mdt   n   m
  ex, ey,ez
Because of symmetry considerations the function
in the integral must be symmetric.
Since the dipole moment is always anti-symmetric,
in order that a dipole transition will be “allowed”, one of
the wave functions must be symmetric and
the other anti-symmetric.
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Line width
Natural line broadening
If the excited state is populated in excess of its Boltzmann population, the excited state will decay
to a lower state until the Boltzmann population is regained. The process is
dNn

 kN n
dt
described by:
when k is the first-order rate constant and 1/k=t.
t is the time taken for Nn to fall to 1/e of its initial value and
is referred to as the lifetime of the state n. If spontaneous emission is the
only decay process than k=Anm.
The Heisenberg uncertainty principle relates the life time to the energy width
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t E   .
From all the above one can show that:
A nm 
64 4 3
4  0 3hc
3
R
nm 2
And from the uncertainty relation we obtain:
 
32 3 3
4  0 3hc
3
R
nm 2
Hence, the linwidth dependence on  indicate that it increases
very fast with the energy. This natural linewidth in typically very
small compared to other sources for line broadening.
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Doppler broadening
The relation of the transition frequency to the velocity of the absorbance relative
 va 
to the radiation source is given by:  a   1  c 


1
where c is the speed of light.
Because there is a Maxwell distribution values of velocities for a given temperature,
the characteristic broadening is given by:
  2kT ln 2 
  
c
m


1
2
where m is the mass of the absorbance.
This broadening is inhomogeneous since not all of the absorbance
are absorbing at the same frequency.
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Pressure broadening:
Collisions shorten the lifetime of excited states.
If t is the mean time between collision and each collision results
in a transition between two states,
  2t 
1
the result line broadening is
.
This broadening produces usually Lorenzian line shape,
however for low frequency transitions a non symmetrical line shape may appear.
Laurenzian line shape
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g ( ) 
 2 
  
2
   0    
 2 
2
Coupling of angular momenta
In the case of hydrogen atom the electronic wavefunction has the form:
 (r, ,  )  R n (r )Ym  ( ,  ) . The Ym  functions are known as the angular
wave functions or, because they describe the distribution of  over the
surface of a sphere of radius r, spherical harmonics.
For , the quantum number n=1,2,3…is the main quantum number and ℓ
is the azimuthal quantum number associated with the discrete orbital
angular momentum values, and mℓ is known as the magnetic quantum
number which results from the space quantization or the orbital angular
momentum. These quantum numbers can take the values:
ℓ= 0,1,2 …,(n-1); mℓ=0,±1,±2,….., ±ℓ
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The function Ym  can be factorized further to give
Ym ( ,  )  2  2 m ( ) exp(im )
1
The functions m ( ) are the associated Legendre polynomials
that are independent on Z, the nuclear charge number, and
therefore the same for all one-electron atoms.
Each electron in an atom has two possible kinds of angular
momenta, one due to the orbital motion and the other to its spin
motion. The magnitudes of the angular momenta are:
  112 
and ss  1 2 
1
The total angular momentum of an electron is given by
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j    s when j    s,   s - 1.......  - s .
There are two limits for coupling of angular momenta1. Coupling of the orbital angular momentum and spin of the electron to produce j.
This limits assumes that the coupling between electrons (spin and orbital angular
momenta) is small. It is called the jj coupling approximation and it is useful only
in the case of heavy atoms.
2. Coupling between the all orbital angular momenta of the electrons is strong and
between the spins is also appreciable. This is the Russel-Suanders coupling
approximation and is the most useful one (also called sometimes the LS coupling
scheme).
In this latter case we define for two electrons L  1   2 , 1   2  1,....., 1   2 and
For L=0,1,2,3 the terms o the atoms are labeled S,P,D,F….
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example the 2p13d1 configuration gives rise to P,D, and F terms.
In a similar way S=s1+s2, s1+s2-1, …
. , │s 1 -s2 │.
The value 2S+1 is called the multiplicity and so for S=0, 2S+1=1
it is called singlet and for S=1, 2S+1=3 and the multiplicity is a triplet.
We can now define the “total angular momentum ”which is given by J=L+S.
J is restricted to the values: J=L+S, L+S-1, ….., │L-S │.
Hence if we have two atoms C 1s22s22p13d1 and Si 1s22s22p63s23p13d1
electronic configurations the following terms can be obtained:
(the assignment is
(2S+1)
LJ)
1
P1, 3P0, 3P1, 3P2,1D1,3D1, 3D2, 3D3,1F3,3F2, 3F3, 3F4
The terms relate to electronic states.
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Coupling between two equivalent electrons –
Here one has to take into account the Pauli exclusion principle
that is in danger unless the two electrons have different values
of either mℓ or ms.
For example we consider carbon in the ground configuration
1s22s22p2. We have to consider only the 2p electrons (n=2 ℓ=1).
For one electron we have ℓ1=1 and (mℓ)1=+1,0,-1 and s1=1/2,
(ms)1=+1/2 or -1/2 and similarly for the second electron. The
Pauli exclusion principle requires that the pair of quantum
number mℓ ms cannot simultaneously have the same values for
the two electrons. Hence it can be shown that only three terms
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arise- 1D,3P,1S.
Hund formulated two rules:
1. O the term arising from equivalent electrons, those with the
highest multiplicity lie lowest on energy.
2. Of these, the lowest is that with the highest value of L.
The splitting of a term by spin-orbit interaction is proportional to
J: EJ-EJ-1=AJ
where EJ is the energy corresponding to J.
If A is positive, the component of the smallest value of J lies
lowest in energy and the multiplet is said to be normal.
If A is negative the multiplet is inverted.
1. Normal multiplets arise from equivalent electrons when
a partially filled orbital is less than half full.
2. Inverted multiplets arise from equivalent electrons when
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a partially filled orbital is more than half full.
Classification of electronic states and
selection rules.
Electronic Spectroscopy of Diatomic Molecules
Molecular orbitals
In the molecular orbitals (MO) approach is to consider the nuclei, without their electrons,
at a distance apart which equal to the internuclear equilibrium distance, and to construct
MOs around them from linear combination of the atomic orbitals (AO).
Electrons are then fed into the MOs in pairs.
Hence the molecular orbital  = ∑i cici when ci are the atomic orbitals.
For the MOs to be different from the atomic orbitals, three conditions must exist:
1. The energies of the AOs must be comparable
2. The AOs should overlap as much as possible
3. The AOs must have the same symmetry properties with respect to certain symmetry
element of the molecule.
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For homonuclear diatomic (lets take N2 as an example) = c1c1+ c2c2.
The 1s AOs satisfy condition (1), since their energies are identical, but not condition
(2).
On the other hand, the 2s AOs satisfy all conditions.
However if we take 2s AO with 2px (z is the axis connecting the two atoms)
the overlap between the two AOs cancel.
From two s AOs it is possible to form two MO with cylindrical symmetry
in respect to the internuclei axis. This will be a s MO.
Two pz AOs also form a s MO. Two px or py AO form  MOs.
When two identical atoms interact it can be shown that two identical AOs
will form two MOs of the form:  
1
 2 2 c1  c 2 
and their energies will be E±= Ec±b when
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The function with the + sign is symmetric and its energy is lower than
the energy of the antisymmetric wavefunction by 2b
(since the two atom attract each other and therefore b is negative).
Symmetric and antisymmetric functions, in respect to the inversion
through the center of the molecule, are assigned as g and u symmetries respectively.
It is common to note s s* and  * for bonding and antibonding characters
of the MOs.
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For example, the ground configuration of the fourteen-electron
nitrogen molecule is:
s g1s 
2
 s g 2s 
* 2
s u 1s
2

2
*
s u 2s  u 2p 4
s g 2p2
Note that bonding  orbitals have u symmetry, while bonding s
orbitals have g.
There is a general rule that the bonding character of an electron in a
bonding orbital is approximately cancelled by the antibonding
character of an electron in an abtibonding orbital.
Therefore we define:
Bond order=1/2 net number of bonding electrons.
Hence,
for nitrogen the bond order is three.
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For heteronuclear diatomic, when the two atoms are similar (NO,
CO, CN etc.) the treatment is the same. When the atoms are very
different (HCl for example) the MO method can be applied but
than because of the difference in energies between the coupled
AOs, the parameter b will usually be very small and therefore the
mixing between the AOs will be small. The measure for the
energy is the ionization potential from each AO.
For example in HCl the chlorine atom has configuration that is
KL3s23p5, but only the 3p electrons have comparable energy with
the hydrogen 1s electon (ionization energies of 12.967 and 13.598
eV respectively). Of the 3p orbitals it is only 3p z which has the
correct symmetry for linear combination with the hydrogen 1s
orbital.
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Classification of electronic states
For all diatomic the coupling approximation that best describes electronic states is
thatwhich is analogous to the Russel-Saunders approximation in atoms (LS). The orbital
angular momenta of all the electrons are coupled to give the resultant L and all the
electron spin momenta produce S.
However, unless one of the nuclei is highly charged, the coupling between L and S is
sufficiently weak, that instead of being coupled to each other, they couple instead to the
electrostatic field produced by the two nuclei. This situation is called Hund ’s case a.
The vector L is so strongly coupled to the electrostatic field and the consequence
frequency of precession about the internuclear axis is so high that the magnitude of L is
not defined, L is not a “good ” quantum number. Only the projection of L on the
internuclear axis,
 , is defined, where
=0,1,2,3. All electronic states with >0 are
doubly degenrate (clock wise and anti clockwise rotation). The value of  are sympolized
by S,P,,F,G …corrsponfing to =0,1,2,3,4..
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The coupling of S to the internuclear axis is due to the magnetic field along the axis due to
the orbital motion of the electrons. The components of S along the internuclear axis is
S  . The quantum number S is analogous to Ms in an atom and can take the values:
S=S,S-1,…,-S.
S remains a good quantum number and for states with >0, there are 2S+1 components
corresponding to the number if values S can take. So for example a state can be 3P.
The components of the total angular momentum along the internuclear axis is  .
  S
For example for =1 and S=1,0,-1 the three components of 3P have =2,1,0 and are
symbolized by: 3P2, 3P1, 3P0. Spin-orbit interaction splits the components so that the
energy level after interaction is shifted by
E  AS
where A is the spin-orbit
couplog constant.
If one of the nuclei is heavy (high Z) LS coupling may be strong enough to couple L and S
to J (Hund’s case c). Now the states will be labeled according to the value of .
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Selection rules
For atoms the selection rules were specified entirely by L,S and J. In diatomic the
quantum numbers , S and  are not sufficient. We must use one (for
heteronuclear) or two (for homonuclear) symmetry properties of the electronic wave
function. [There is a relation between the number degree of freedom and the number
of quantum numbers required to describe the system].
The first symmetry is g or u, which indicates that e is symmetric or antisymmetric
respectively to inversion through the center of the molecule (this is for homonuclear
molecules only). The second element of symmetry applies to all diatomic and
concerns the symmetry of e with respect to reflection across any (sv) plane
containing the internuclear axis, If e is symmetric, the state is labeled + and if it is
antisymmetric it is labeled -. This is usually used for S states. In P states there is
double degeneracy due to the + and – states but the label is not often used.
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Since we consider electric dipole type electronic transitions
the selection rules are:
  0,1 for example S - S, P - S,  - P transiti ons are allowed
1.
but not  - S or F - P.
2. S=0. This restriction does not hold for very heavy atoms,
for example in I2.
3. S=0; =0,±1 for transitions between multiplet components.
4.
  ;  
This relevant only for S-S transitions so that only
S+-S+ or S--S- are allowed.
5. g«u; No g to g or u to u transitions.
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