Transcript Document 7249461

Electronic Spectroscopy
Why Electronic Spectroscopy?
• Gives information on electronic structure
• Shorter wavelengths allow tighter focusing
and thus imaging. (~l/2n).
• Can be used to instigate photochemistry
• High photon energy means negligible thermal
background except at very high T.
– We can “count” individual photons with high
detection efficiency using PMT’s & APD’s.
Why (cont)
• High emission rates (up to ~109 per second) mean that
single atom or molecule is potentially visible.
– Single molecule spectroscopy has exploded in popularity in
past decade.
– Must be photochemically stable.
• Quenching and energy transfer (FRET) important
problems and also useful probes of structure.
– GFP’s and nanodots are now widely used at fluorescence
“tags.”
• With ionization, we can mass select and detect with
100% efficiency.
• Ultrashort laser pulses allow for one to follow very fast
molecular dynamics.
Experimental Methods
• Vis/UV absorption
• Emission Spectroscopy
– Grating spectrograph (CCD cameras)
– FT instruments.
• Supersonic jets used to reduce spectral congestion and simplify spectra.
Double resonance methods also used.
• Laser Induced Florescence.
– Use of collimated molecular beam can dramatically reduce Doppler
broadening.
– Selective mobility methods increasing used to separate.
• Resonant Enhanced Multiphoton Ionization (REMPI) – often combined
with time of flight.
• Photoacoustic or optothermal for emitting states.
• Photoelectron spectroscopy (often with He I lamp which produces 21.4 eV
photons).
Hartree-Fock Theory
• Wavefunctions approximated by single Slater
determinant |fafbfc….| where fa are are set of
orthonormal spin orbitals
• The are eigenfunctions of Fock Operator
– Ffa = eafa
• Where F includes interaction of electrons with
the average density of all the other electrons plus
an “exchange” term.
• HF wavefunction gives the lowest possible energy
for a single determinant form
HF-cont.
• The set of all SD formed from all HF spin orbitals span a
complete set of symmetry allowed (antisymmetric)
many electron wavefunctions.
• The energy required to ionize by removing an electron
from the fa orbital is just -ea (Koopman’s theorem)
– Cancelation of lost correlation energy and reorganization
energy.
• The energy to promote an electron from a filled to
empty orbital fa -> fb is NOT eb  ea.
– The orbital energy for empty orbital includes interaction
with electron that left fa orbital
• The fa are usually expended in terms of atomic basis
functions; today Gaussian bases sets most often used.
HF Selection Rules
• Only allowed optical transitions allowed in HF are between states
whose determinants differ by a single spin orbital --- one electron
rule
– The product of the initial and final spin orbitals must transform as a
component of the dipole moment in Point Group of molecule.
• Total product of total electronic symmetry of initial and final states
must also transform as dipole component.
• Total electron spin S is a good quantum number for HF
wavefunctions.
– For partially filled orbitals, we often need to take linear combinations
of determinants to produce a spin Eigenstate.
– For both electric and magnetic dipole transitions, only DS = 0
transitions allowed.
– Spin orbit coupling “spoils” S as a quantum number and leads to
nonzero intensity for transitions that violate DS = 0 rule.
– Electric dipole transitions only allow transitions where spin projection
does not change, magnetic dipole transitions allow changes in the spin
projection.
Diatomic Molecules
• Projection of total Electronic orbital angular
momentum (L) is good quantum number.
– L = 0, ±1,±2… producing S, P, D, etc. states.
• For L ≠ 0, have B field along bond (z), S
precesses around z to give quantum number
S. W = L + S, total angular momentum
number along z. R (perpendicular to z) is endover-end rotation. J = R + Wz. (Case A).
– Spin orbit coupling A L S. A can be
approximated by atomic z. (pure precession
model)
Diatomic (cont)
• We used lower case s, p, d, etc. for projections of
orbital functions.
– Splitting pattern of atomic orbitals largely reflects bonding,
nonbonding, and antibonding character of overlap.
• For S states we have S+ and S-, giving symmetry with
respect to reflection in plane that includes z axis.
– State (such as O2) with 1e in each of p orbitals gives a Sstate.
• For homonuclear diatomics, we add g/u label to
indicate if symmetric or antisymmetric with respect to
reflection in reflection symmetry plane perpendicular
to z axis.
Diatomic e-Dipole Selection rules
• DL = 0 (parallel) or ±1 (perpendicular).
• DS = 0 (neglecting spin orbit coupling).
– DS = 0 for case A
• DW = 0, ±1
• S+ <-> S+ , S- <-> S- but not S+ <-> S• g <-> u but not g <-> g or u <-> u (allowed by magnetic
dipole).
• In HF, we still have one electron rule!
– Change in projection for excited electron limited to 0, ±1.
– Due to configuration interaction, transitions that violate
one electron rule gain intensity but are generally weak.
Vibrational Structure
• The radial potential energy curves different for
each electronic state. In general, each
vibrational state of one state can couple to all
those of any other:
M e (v',v") =
 '
v'
(r)m(r)v'' (r)dr
m(r) =   ' me  ' d n re
e
e
• Often neglect radial dependence of m(r)

M e (v',v")  m(r0 ) f v',v"
Fv',v" = f v2',v"
F
v',v"
v'
"Franck  Condon factor"
=  Fv',v" = 1
v"
f v',v" = v'  v"
Franck-Condon
• If we continuum vibrational functions, part of
sum becomes integral over FC density.
Continuum functions must be normalized with
respect to integration variable.
• We can improve upon simple FC
approximation if we evaluate electronic
transition dipole at r-centroid:
v' r "
r (v',v") =
v' "
Franck-Condon (cont)
• If we neglect change in vibrational frequency
and consider displacement, then FC from
ground vibrational state of either surface
follow Poison distribution.
sv s
2
F
=
e
s

DR
0,v
e
• Note that by fitting
v!
the FC intensities, we can determine the
magnitude of DRe but not its sign.
 in vibrational frequency
-- Usually, the change
opposite in sign to change in Re.
Franck-Condon (cont II)
• For given initial state, largest FC factors are to
vibrational states of other surface that have
similar inner or outer turning points.
– For bound to free transitions, this applies to
energy corresponding to peak of continuous
absorption spectrum.
• The nodal structure in the initial wavefunction
are reflected in number of sign changes of fv’v”.
Deslandres Tables
• Matrix of v’, v” with band origins in each entry.
– Differences between elements in each pair of columns
and each pair of rows should be constant –
combination differences.
• The strongest transitions follow parabola running
through table.
• Often, absolute quantum number assignments
can be ambiguous – i.e. lowest observed state
may not be v = 0 – isotopic substitution will
generally resolve this.
Rotational Structure (Singlet-Singlet)
• Usual DJ = 0, ±1 selection rule. Minimum J = W.
– For SS transition, no DJ = 0
• P and R branch transition wavenumbers fit standard
polynomial expression in transition number m.
– Unlike vibrational case, DB = B’- B” is often quite large.
– This leads to a “band head” where transition wavenumber
vs. m has maximum in R branch (for DB < 0) or minimum in
P branch (DB < 0)
 head
(B'+B") 2
= 0 
4(B' B")
– The first is known as red-degraded band, the second bluedegraded.

– See figures 9.14 and 15 of Bernath showing band heads at
low resolution and also a Fortrat diagram showing
wavenumber vs. m number.
Rotation-non singlet transition
• If L = 0, then spin weakly coupled to molecular axis. Instead,
end over end rotation of molecule creates magnetic field
parallel to N. S precesses around N to produce J. (Case B).
– Leads to spin-rotation splitting of lines, usually small compare to B
value.
• If both L and S not zero, we have to consider their coupling
due to spin orbit.
– As molecule rotates faster, S cannot rotate around moving axis. Leads
to “uncoupling” and transition from case A or case B coupling cases.
– Various possible cases treated in Detail in Herzberg’s diatomic
molecule spectroscopy book.
• When molecules produced by photodissociation, the relative
populations in different states gives information on transition
state, for example if the unpair electron was in an in-plane or
out-of-plane orbital.
Dissociation and Predissociation
• If there is a large change in Re, we can have FC factors up to
highest bound level and beyond producing a continuum
spectrum.
– Fitting of highest vibrational term values gives very precise
dissociation energies.
– Due to centrifugal barrier at higher J values, the highest
vibration state(s) can be quasi-bound and show broadening due
to tunneling through barrier.
• Excited electronic states are often crossed by repulsive
states (see fig. 9.28). If these are of the same symmetry,
there will be an avoided crossing. Even if of different
symmetry, rotation with often couple the surfaces. On
each vibration, the excited state can “cross over” and
“predissociate”.
– Often detected as break-off of fluorescence intensity.
– By measuring kinetic energy of fragments, using ion-imaging
methods, precise bond dissociation energies can be determined.
Electronic Spectra of Polyatomics - Notation
• For most molecules, the ground state has all the electrons paired in
the occupied orbitals and thus is a singlet, labeled S0. The low lying
excited singlet states are labeled S1, S2, …. Involve promotion of one
electron from an occupied orbital to unoccupied orbital. The
lowest is the promotion from HOMO -> LUMO.
• For each excited singlet, there is a triplet with the electrons in the
same orbitals but with the spins parallel instead of anti-parallel.
These are labeled T1, T2,…. Each triplet state is typically ~0.5-1 eV
bellow the corresponding singlet.
• We label the vibrational transitions only by listing the normal
modes involved in a transition. 1231 is a transition from the ground
vibrational state to one with two quanta in mode 1, one quanta in
mode 3 and no quanta in any other modes. If we have a mode
excited in the ground ground state, we indicate the number of
quanta in that mode by a subscript. The transition labeled 1231411 is
what we call a “hot band” because the mode 4 starts excited “but
just goes along for the ride.” Note that it will (in the harmonic
approximation) be shifted from 1231 by the difference in
wavenumber of mode 4 between the states.
Polyatomic Molecules
• Franck-Condon Factors
– If the normal modes of the two electronics are the
same, expect for a displacement, then the FC factors
are simple products of the FC factors for each mode,
which can be calculated easily
– Usually, only a few modes have significant FC
displacements and so far from the full 3N-6 vibrational
modes show vibrational progressions.
– Only modes that are totally symmetric in the group of
common symmetry elements of the two electronic
states can have displacements.
• Note that the molecular symmetry will often change upon
electronic excitation!
Duchinsky Rotation
• In general, the normal modes of the upper state
are linear combinations of those of the lower
state: Q’ = S Q” + d
– Where S is a matrix that can be written in terms of the
product of the l matrix of one state times the
transpose of the l matrix for the other.
– In general, S will block diagonalize in terms of irreps of
common symmetry group
– There exist a set of recursion relationships that allow
one to efficiently calculate all the FC factors. Doktorov
et al. J. Mol. Spec. 139, pg 147-162 (1977).
Vibronic (Herzberg-Teller) Coupling
• For molecules with high symmetry, many
electronic transitions are forbidden. For example,
in the case of benzene, the first two excited single
states are forbidden from the ground state.
• However, it is always the case that there will be
modes that distort the molecule in such a way
that the transition is not longer symmetry
forbidden.
 m 
• We can write to first order m   Q Q
e
e
k

k

k
Vibronic Coupling (cont).
•
If we have such a mode k, the 0-0 transition (between ground vibrational states in
both electronic states) will be forbidden, but the transition to k1 (one quanta in
mode k) will be allowed.
– In general, we will have FC progressions in the totally symmetric modes “built” upon this
“false origin” T0 + nk’. Bernath shows (Fig. 10.12) such a progression built on mode 6 in
benzene’s S0->S1 transition.
– In emission from the ground vibrational state, we will excite mode k in the ground state and
have false origin of T0 – nk”.
– In general, we have many vibronically active modes, but sometimes, one dominates.
•
•
•
To be vibronically active in absorption form the ground state, the vibronic
symmetry of a state (product of electronic and vibrational symmetry) must have
same symmetry as one of the components of the dipole.
Sometimes a transition is allowed but very weak. In those cases, vibronic effects
can play an important role in the spectrum. This is often seen my the direction of
the transition dipole moment being different for different vibronic bands in the
same electronic transition.
In the “crude” BO approximation, we neglect changes in the electronic
wavefunction with vibration. In that case, the vibronic coupling is expressed in
terms of mixing of different electronic states by vibration.
Jahn-Teller Effect
• Jahn and Teller proved a theorem that for any
nonlinear molecule, any degenerate electronic
state will be unstable to distortion that will
eliminate the degeneracy.
– Most common is the “e x E” Jahn-Teller case
where double degenerate (E) electronic state
distorts along a doubly degenerate mode (e).
– Case in point is O3, which has asymmetric
structure. In a D3h configuration, we would have
an E symmetry state.
e x E “Mexican Hat” potential for linear
Jahn-Teller coupling
When quadratic coupling terms are included, we get 3 equivalent minima in the trough.
Motion around the trough is known as pseudo-rotation as each atom moves in a circle
But with different phase.
Berry’s Phase
• If we move adiabatically around the trough of the
Mexican hat, the electronic wavefunction returns to
where it started changed in sign. The overall
wavefunction cannot change sign, so this means the
vibrational part must.
– This implies that the pseudo-rotation is characterized by ½
integer quantum numbers.
• This was first worked out by Herzberg and LonguetHiggins but later rediscovered in a much more general
context by Michael Berry and is now known as a
molecular application of Berry’s phase.
Nonradiative Decay
Internal Conversion
• At energy of S1, the vibrational states of S0 are usually
extremely high – formed a quasi-continuum (or real
continuum if IR radiative width is included).
• States are coupled by terms such as:

a
  "e   "vib 
 


 Qa  Qa 
'
e
'
vib
• The matrix elements are normally small because
vibronic states of the same energy differ by many

vibrational
quanta leading to interference.
– When excited state come close in energy, then the
coupling can be large.
Kasha’s rule
• In condensed phase, we usually have vibrational
relaxation rates that are much faster than
radiative so after excitation, molecules cool their
ground vibrational states.
• In most cases, the rate of Internal conversion
between excited singlet states is also fast because
these states are close in energy
• Kasha’s rule is that regardless of the excitation
energy, the fluorescence is always from S1 and
the phosphorescence from T1.
Conical Intersections
• Electronic states of different symmetry can cross,
but only as long as there is no displacement in
modes that reduce them to the same symmetry.
• When this happens, we can have extremely fast
nonadiabatic transfer between electronic
surfaces.
• Even for surface of same symmetry, there are
seams of conical intersections in 3N-8
dimensions.
PALM/STORM
• Image through a diffraction limited microscope onto high-res CCD
camera
• Label compound of interest with compound that does not fluoresce
but can be photolyzed to produce a efficient fluorescent dye.
• Photolyze weakly with UV to produce an array of resolved single
molecules in image.
• Follow the emission of individual molecules. In principle, one can
determine the center of the ~l/2 wide image of each molecule to
about N-1/2 where N is the number of photons detected before dye
photochemically bleaches. ~10 nm resolution achieved in practice.
• Continuously produce new chromophores at rate at which old ones
bleach so that one continuously observes single molecules in the
image.
• Produce place “dot” at center of each observed diffraction limited
spot.
• If one focuses above and below the sample plane, then the image
spot size can be fitted to determine the “z” position of the emitter
with only modest degradation of x,y resolution.
Stimulated Emission Depletion microscopy (STED)
Stephan Hell
• Focus excitation laser (TEM00) and “de-excitation”
laser (TEM10 doughnut mode) on same spot.
Emission will only come from near the center hole of
TEM10 mode.
STED