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 end-over-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 S- state. • 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: • Often neglect radial dependence of m(r)
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:
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. • Note that by fitting the FC intensities, we can determine the magnitude of DRe but not its sign. -- Usually, the change in vibrational frequency 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) • The first is known as red-degraded band, the second blue-degraded. • 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. • We can write to first order
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 Longuet-Higgins but later rediscovered in a much more general context by Michael Berry and is now known as a molecular application of Berry’s phase.
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: • 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.