28/08/2013 Molecular spectroscopy DREAMS Laboratory Scuola Normale Superiore, Pisa ELECTRONIC VIBRATIONAL ROTATIONAL Malgorzata Biczysko E Experimental and computational spectroscopy Experimental and computational spectroscopy Unified and general theoretical model Virtual spectrometer = vis-à-vis comparison • From 1 vibrational level to • From theory to spectra transitions between 2 PES • Direct comparison with experiment – • Accuracy interpretation Vibrational averaged properties • Easy-to-use procedure + analysis tools – Vibrational spectroscopies – Vibronic transitions • Focus on nuclear part – Beyond vertical transitions and harmonic approximation • General model – Many spectroscopies – the same tool – Easy extensions – common kernels 1. V. Barone, J. Bloino, M. Biczysko, F. Santoro, J. Chem. Theory Comput., 5, 540 2009; JCTC, 6, 1256 2010 2. VB, A. Baiardi, MB, J. Bloino, C. Cappelli , F. Lipparini , Phys. Chem. Chem. Phys, DOI: 10.1039/C2CP41006K, 2012 1 28/08/2013 Experimental and computational spectroscopy Theoretical models: PES Overview Global Potential Energy Surfaces • Vibronic spectroscopy Stationary Full PES representation points – General formulation for one-photon processes – The electronic transition integral – Calculation of the band-shape – Approximation schemes • Anharmonic vibrational spectroscopy – Vibrational energy levels – Transition intensities – Vibrational averages of arbitrary properties – Medium-to-large systems Global potential surfaces for N2H2 Theoretical models: PES Theoretical models: PES PES representation LOCAL Potential Energy Surfaces dimension • Numerical grid representation 1-D Full-D H1 – Assuming 10 points/coordinate Full 3-D PES for HBS, HBN 10 N X ... Dimension H H3 Number of atoms Grid 180 3.2 3 H2 3 3 10 =1000 3 2.8 6 0 4 6 10 =1'000'000 ! 1 2 2.6 R(BH)/bohr (3n-6) 1-D PES for proton motion in XH-NH 2.4 n 3n-6 10 ! ! 3 2.2 • Analytic function 2 1.8 – Eg. 6-D global PES for water dimer (X. Huang et al. J. Phys. Chem 1.6 A. 2005, 110, 445) 2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.2 R(BS)/bohr – fifteen variables of the fit, the polynomial to seventh order 3-D PES (NHsymm, NHasymm, N-Ni) for NH3@Ni(111) – in total there are 5227 terms in the expression for the PES 2 28/08/2013 Theoretical models: PES + PS Theoretical models Spectroscopy: PES + Property Surface Sampling of PES and PS • PES: dynamics, structures Time-independent Time-dependent – In principle can be studied by fully classical methodologies (Force Fields) + molecular mechanics or classical dynamics • PES + PS : spectroscopic properties – Interaction with light – intrinsically quantum effect, electronic structure need to be taken into account – ‘spectroscopically relavant part’ need to be computed at QM level Eigenvalues, eigenfunctions wavepackets propagation • Large systems time-dependent correlation functions – QM/QM’ (QM’: semiempirical) or QM/MM Search for stationary points: minima (determination of structure and properties), Sampling of the energy surface: collection of • Example saddle points (determination of energy trajectories (Classical or Quantum Molecular – Design of laser pulses for mode specific barriers and reactivity) Dynamics approach) vibrational excitation in an enzyme–substrate PES and PS described derivatives wrt complex PES and PD are determined ‘on the fly’ nuclear coordinates (eg. Normal modes) – QM/MM PES and dipole moment PS – Chemical Physics Letters 491 (2010) 230–236 Or a priori from grid calculation Theoretical models: PES Theoretical models: PES Time-independent + feasible approaches Coordinate system • Born-Oppenheimer approximation • Vibration described with respect to the center of mass – One excited electronic state at a time – Periodic, concerted displacement of nuclei (independent excited states = no coupling) – center of mass unaltered in laboratory space • Harmonic description of PES as a 1st • N-atomic molecule: 3N coordinates order approximation – x,y,z for each atom • Body-fixed coordinate system – Semi-rigid molecular systems – Origin (center of mass) wrt Laboratory space : 3 coordinates (translation) – PES close to the local minimum – Space orientation: 3 rotational coordinates , , - Eulerian angles – Analytical (GS harm) or numerical (ES • 3N-6 vibrational coordinates (linear molecules 3N-5) Rotational harm, GS anh) coordinates – Anharmonic corrections via perturbation y theory • Normal modes representation – Based on Cartesian coordinates y • Eckart conditions – Separate translation from x rotation/vibration Center of mass – Minimize interaction between rotational z x and vibrational motions z Displacement vectors Laboratory space 3 28/08/2013 Vibronic spectra: Theoretical framework Electronic spectroscopy One-photon processes • General sum-over-states formula to compute spectrum band shape 1. Transitions between diferent PES 2. Computation of VE and transition moments 3. Vibrational effects 4. Simulation of spectra line-shape 5. Approximations – accuracy 6. Examples power of incident energy lower state higher state distribution function constant Boltzman population Vibronic spectroscopy: Theoretical framework Vibronic spectra: Theoretical framework Equivalency table for the properties The electronic transition integral electronic nuclear – Only the vibrational wave function must be considered – New transition dipole moment integral: 1. J. Bloino, M. Biczysko, F. Santoro, V. Barone, JCTC, 6, 1256 (2010) 4 28/08/2013 Vibronic spectra: Theoretical framework Vibronic spectra: Theoretical framework The transition dipole moment Mathematical formalism of FC approximation • There is no general analytical solution for • Approximation based on the Franck-Condon principle – Short-time electron jump nuclei are nearly unaltered by the molecular vibrations – Franck-Condon: is constant during the transition – Herzberg-Teller: varies linearly with the normal coordinates • The problem is reduced to calculating overlap integrals – Taylor expansion of about the between the vibrational wave functions v v equilibrium geometry of one electronic • But r(m) and s(n) are expressed with respect to different state basis sets, Q’ and Q’’, respectively! Vibronic spectra: Theoretical framework Vibronic spectra: Theoretical framework Transformation between the sets of coordinates General case: calculation of • The normal modes in S and S do not coincide (J≠I) • Presence of Hermite polynomials Hv : complex analytic 0 1 formulae • Analytic formulae – One equation per overlap integral • ad hoc expressions for each overlap integral • But all necessary transitions must be predicted before actual calculations – General loops for a given number of excited modes K • More general-purpose than per-integral equations • But loop nesting depends on highest number of excited modes supported – Complexity of expressions grows quickly with number of quanta • Good approximation if no large distortion during the transition • Recursive approach – Only 0’|0’’ must be known exactly • Other approaches – Two formulae are needed to compute any Franck-Condon integral – Curvilinear transformation [Lucas, N., J. Phys. B, 6, 155 (1973)] – But storage required to avoid redundant calculations – Cartesian coordinates 5 28/08/2013 Vibronic spectra: Theoretical framework Theoretical framework Calculation of : Treatment by classes To deal with infinity: selection of transitions • Definition (class) • Recursive formulae – – possible to evaluate the transition dipole moment integrals A class represents the number of simultaneously excited by evaluating the overlap integrals between the vibrational initial and final modes in a state C1 and C2 intensities, states (FC integrals) computationally cheap – Ex.: |10i in class 1, |2i ; 4j ; 8k in class 3 • In principle, – there is an infinite number of transitions to evaluate • Recursion and analytic Oscillators of final state – But, most transitions have a low-to-negligible probability formulae 3 – For classes 1 and 2, per- 2 C1 C2 integral analytic 1 … |0’’ 0 v’’ formulae are used when 1 v’’2 v’’n 0|0 available, recursion C3 formulae otherwise 3 – For classes 3 and higher, 2 … recursion formulae are 1 0 |0’ v’ used 1 v’2 v’n Oscillators of initial state Vibronic spectroscopy: sum-over-states Vibronic spectroscopy: theoretical fremework A priori selection of transitions Sum rules and prescreening accuracy • Setting • Screening is used to select – The selection of transitions to relevant transitions among compute induces an approximation to the generated spectrum an infinite number of – The extent of the approximation for a combinations priori schemes is generally not known – It is possible to know analytically the • a priori screening total intensity using sum rules – avoids unnecessary calculations but transition integrals must be estimated • approach has been designed to be interfaced with various schemes 2 • current default prescreening: FCClasses1 0K 0' if '' – Transitions from classes 1 and 2 calculated up to a chosen limit and used to spectrum progression collect data: tot I Checked after each class! – For each class starting from 3, the highest number of quanta for each mode is calculated to stay below predefined number of integrals max • The calculations depend on the number of integrals ( N I ) to compute NOT on the dimensionality of the system 1. F. Santoro et al., J. Chem. Phys. 126, 169903 (2007) and 128, 224311 (2008) 6 28/08/2013 Validation: a priori selection scheme Validation: a priori selection scheme Absorption spectrum of Coumarin 153 Phosphorescence spectrum of Chlorophyll c2