The spectral method: time-dependent quantum dynamics of FHF - : Potential Energy Surface, Vibrational Eigenfunctions and Infrared Spectrum. Guillermo Pérez Hernández Friedrich Schiller Universität Jena Institut für Physikalische Chemie 3 December 2007, Jena

2 OUTLINE - FHF - - The vibrational problem - Theoretical approaches - Time dependent approach: the Spectral Method - Results - Quantum Chemistry - Nuclear Dynamics - Spectra - Conclusions and outloook

3 FHF - - Bifluoride Ion: - d F-F ~ Å - 20 e (closed shell ground state  g.). - Heavy-light-heavy system ( Zundel cation, CdH 2 ) Very strong HB ~ Kcal / mol Suitable for high-level ab- initio calculations!

4 The vibrational problem - Normal modes of vibration around the equilibrium geometry - Vibrational degrees of freedom: 4. symmetric stretch asymmetric stretch bending (x2) 3 2 1

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6 The vibrational problem - Vibrations: INFRARED (IR) Spectrum SOLID: B. S. Ault, JPC 83, 837 (1979) GAS: Kawaguchi et al, JCP 87, 6838 (1987) 3 (as. stretch) 2 (bending) KFH 2 in solid Ar matrix

7 vibrational coordinates (q) What do theoretical approaches consist of? Theoretical approaches energy / cm -1 h 2 h 3 electronic ground state TIME INDEPENDENT SCHRÖDINGER EQUATION potential energy surface (PES)

8 vibration coordinates (q) What do theoretical approaches consist of? Theoretical approaches energy / cm -1 h 2 h 3 electronig ground state TIME INDEPENDENT SCHRÖDINGER EQUATION potential energy surface (PES)

9 - The many different approaches differ basically in these aspects: - Quantum Chemistry for calculation of PES. - Methods: (HF, MP2, CISD..., CCSD....) - Basis sets: (augmented, polarized....) - Nuclear Dynamics for obtention of eigenfunctions and eigenvalues. - Selection of coordinates: (normal mode, bond-angle, spherical...) - Representation of space: (on a grid, with analytical functions...) - Representation of PES: (on a grid, fit to analytical eigenfunctions, force fields...) - „Technical“ approximations - Variational procedures mostly involved. - They are time independent. Theoretical approaches

10 - The Spectral Method (Feit et al., J. Comp. P, 41, 112 (1982)). - Obtains time independent information in a time dependent fashion. - Implies solving the time dependent Schrödinger Equation, which has the solution: Time dependent approach: the spectral method initial state state at time t economic, flexible !

11 Time dependent approach: Split-Operator How to apply the time-evolution operator?

12 - The time propagation provides time dependent information about the system: wavefunctions, mean values: 1. Time dependent approach: the spectral method time vib. coord t2t2 t3t3 titi tftf t1t1 t0t0 timewfs exp. values

13 - How do we transform the previous time dependent information into eigenvalues and eigenenergies? Time dependent approach: the spectral method autocorrelation funcion Fourier Transformpower spectrum complete basis set of eigenfunctions

14 - How do we transform the previous time dependent information into eigenvalues and eigenenergies? 2. Time dependent approach: the spectral method autocorrelation funcion Fourier Transformpower spectrum complete basis set of eigenfunctions

15 2.Time dependent approach: the spectral method - The power spectrum displays the eigenenergies. energy spectrum -Eigenvalues are known, but they are only "numbers“. -Still no information about the eigenfunctions themselves. - Which eigenvalue corresponds to which eigenfunction?

16 energy spectrum 3. Time dependent approach: the spectral method - Obtaining the eigenfunctions: the filtering procedure. filtering operation filtered eigenfunction filter value

17 Time dependent approach: the spectral method - Obtaining the eigenfunctions: the filtering procedure. filter value filtering operation filtered eigenfunction

18 RESULTS Quantum Chemistry

19 - Coordinate system - Molecule is assumed to have zero total angular momentum (J=0). X Y Z x y 0. Results: Quantum Chemistry Grid size: R 64 points x 64 points y 64 points 64 3 points! R

20 - PES - Global minimum at x=0, y=0 and R= 2.28 Å (R eq exp. = 2.28 Å) 0. Results: Quantum Chemistry (CCSD(T) / aug-cc-pvtz) R = 2.28 Å X Y R

21 0. Results: Quantum Chemistry (CCSD(T) / aug-cc-pvtz) R = 1.82 ÅR = 3.2 Å - PES

22 RESULTS Nuclear Dynamics

23 - Initial wavefunction displaced from global minium (R≠2.28, x,y≠0). - Total propagation time 32 ps. - Expectation values for the first 200 fs 1. Results: Nuclear dynamics. Propagation. - Autocorrelation function for the first 200 fs.

24 2. Results: Spectrum resolution ~ propagation time 1 resolution = 2 cm -1 prop. time = 33 ps

25 Results - Example of a filter operation: - After relative short integration time, the filtered wavefunction shows its characteristic structure with nodes. t = 0 fs t = 150 fs

26 Movie time

27 3. Results: Eigenfunctions. - Vibrational ground state and fundamental modes: R y x 1 2 3

28 2. Results:Transition frequencies. -Comparison of the obtained transition frequencies with other available theoretical and experimental data:

29 IR Spectra: Modulating intensities Transition probability: permanent dipole moment:  x and  y

30 IR-Spectra: Intensities

31 Conclusions and outlook -Time dependent approach provides very flexible, economic and reliable method. -Good agreement with experimental and other theoretical data. -Outlook: - Take into account the degenerate bending mode - Trigger that motion with a laser?

32 Acknowledgments - Prof. Leticia González - Dr. Jesús González-Vázquez - GK # and you!