Computational SpectroscopyII. ab initio Methodsfrom part (d) Electronic SpectraChemistry 713Updated: February 20, 2008The Born-Oppenheimer Approximation For a given molecular geometry (i.e., fixed nuclearcoordinates, R), solve the electronic Schroedinger equation:where He is the whole molecular Hamiltonian except thenuclear kinetic energy and r represents the coordinates forall of the electrons, and ψe is the electronic wave function.Repeat for a range of molecular geometries R of interest toconstruct a potential energy surface. The electronic energy En(R) is the potential energy inwhich the nuclei move. Up to now we have just been concerned about the lowestenergy electronic state, n=0. To deal with electronic (UV/vis) spectroscopy, we alsoneed to know some of the higher electronic surfaces (n=1,2, …) as well. The nuclear motion on each surface can then be solved as aseparate step.! He"e,nr;R( )= EnR( )"e,nr; R( )F.F. Crim, Spectroscopic probes and vibrational state control of chemical reaction dynamics in gases and liquids.Talk WA04, International Symposium on Molecular Spectroscopy, Columbus OH, 2006.http://molspect.chemistry.ohio-state.edu/symposium_61/symposium/Program/WA.html#WA04The Franck-Condon Principle In a diatomic molecule, the potentialenergy curves are different for lower andupper electronic states. The bond length re changes The vibrational frequency ν changes. Use double prime for lower state (″), andsingle primes for upper state (′).Gordon M. Barrow, An Introduction to Molecular Spectroscopy,McGraw-Hill, New York, 1962, fig. 10-1, p. 232.rre′re″hν′hν″Iodine oxide (IO)Potential energy curves There are many potential energy curves even in asmall molecule. Some are attractive; others are repulsive Curves of the same symmetry don’t cross: “adiabatic” curves Some result from the crossing of “diabatic” curves, andas a result have peculiar shapes. Notation: “X” denotes the ground state Upper case letters, A, B, etc., indicate excited states ofthe same spin multiplicity as the ground state. Lower case letters, a, b, c, etc., indicate excited states of adifferent multiplicity. (Numbers are not normally used.) The symmetry and spin multiplicity of the state areindicated by a term symbol, such as 2Π, 4Σ–, etc.S. Roszak, M. Krauss, A. B. Alekseyev, H.-P. Liebermann, and R. J. Buenke, J. Phys. Chem. A, 104 (13), 2999 -3003, 2000. 10.1021/jp994002lr, Fig. 1.The Franck-Condon Principle Electronic transitions are “vertical”, that is the nucleidon’t move while the electron(s) are being excited. Because the upper state wavefunctions are shifted fromthose in the ground state and because the vibrationalfrequencies are different, changes in the vibrationalquantum number accompany the electronic excitation. The relative intensities of the vibrational subbandsv′←v″ are given by the squares of overlap integrals,called Franck-Condon factors: If neither the bond length, nor the vibrational frequencychange, then the selection rules are Δv=0. In polyatomic molecules, vibrational progressions occurin vibrational modes for which either the equilibriumposition is changes or the frequency is changed.! "# v "# # v 2 In polyatomic molecules, vibrationalprogressions occur in vibrational modes forwhich either the equilibrium position ischanges or the frequency is changed. Therefore, a typical electronic band has a lot ofvibrational structure, which extends over a fewthousand cm-1. The band origin is the frequency of the v=0←0 band. The band origins of electronictransitions are what we can most easilycalculate with ab initio methods. For large molecules or in the condensed phase,the vibrational structure is heavily overlappedand merges together into a wide unstructuredblob (the Franck-Condon envelope).The Franck-Condon Principle:Polyatomic moleculesA spectrum with vibrational progressions45,000 37,000Wavenumber / cm-1BandOrigin Franck-Condonenvelope BenzeneJ. M. Hollas, High resolution Spectroscopy, Butterworths, London, 1982, p 393.Selection rulesfor electronic spectroscopy Spin multiplicity is conserved. Changes in vibrational motion follow the Franck-Condon Principle Rotational transitions (ΔJ=0,±1, ΔK=0,±1)accompany each electronic+vibrational (vibronic)transition. For molecules with a center of symmetry, the g/usymmetry changes. Nuclear spin states are conserved. Additional rules apply in particular cases.The fateof electronically excited molecules1. Fluorescence: a visible or UV photon is emittedto return the molecule to its ground state.2. Intersystem crossing: radiationless conversion ofthe energy back to a state of different spinmultiplicity. (e.g., singlet to triplet).- Occasionally followed by phosfluorescence:emission of a photon with a change in the spinmultiplicity. (VERY weak; a long radiativelifetime.)3. Internal conversion: radiationless conversion ofthe energy back to the ground state (or other stateof the same spin multiplicity).4. A Photochemical reaction- photodissociation, isomerization5. Energy transfer to a nearby moleculeJablonski diagramJ. I. Steinfeld, Molecules and Radiation, MIT Press, Cambridge, MA, 2nd ed, 1985, p 287.Conical intersections Two electronic surfaces canmet like two cones touchingtip to tip. Widespread throughoutelectronic spectroscopy. Act like a sink-hole thatallows the system to dropthrough onto a lower surface. Action spectra can be recordedby detecting photofragments. Note that only two of the sixvibrational coordinates arerepresented in this diagram!Conical intersectionhνF.F. Crim, Spectroscopic probes and vibrational state control ofchemical reaction dynamics in gases and liquids. Talk WA04,International Symposium on Molecular Spectroscopy, ColumbusOH, 2006.Electronic excitationsin the orbital approximation For electronically excited states, one or more electrons is in an orbital withhigher than the lowest possible energy allowed by the Pauli principle. Given M doubly occupied molecular orbitals, and N unoccupied orbitals(N→∞), there are an enormous number of possible excited electronic states. Consider cases where the ground state is closed shell, and can be representedby a single Slater determinant: An electron in the highest occupied molecular orbital (HOMO) ψN/2 is excitedto a higher orbital, ψa:A singly occupied orbital with no bar is spin up; one