High order efficient splittings for the semiclassical time-dependent Schrodinger equation

[EN] Standard numerical schemes with time-step h deteriorate (e.g. like epsilon(-2)h(2)) in the presence of a small semiclassical parameters in the time-dependent Schrodinger equation. The recently introduced semiclassical splitting was shown to be of order O (epsilon h(2)). We present now an algorithm that is of order O (epsilon h(7)+epsilon(2)h(6)+epsilon(3)h(4)) at the expense of roughly three times the computational effort of the semiclassical splitting and another that is of order O (epsilon h(6)+epsilon(2)h(4)) at the same expense of the computational effort of the semiclassical splitting.The work of SB has been funded by Ministerio de Economia, Industria y Competitividad (Spain) through project MTM2016-77660-P (AEI/FEDER, UE).Blanes Zamora, S.; Gradinaru, V. (2020). High order efficient splittings for the semiclassical time-dependent Schrodinger equation. Journal of Computational Physics. 405:1-13. https://doi.org/10.1016/j.jcp.2019.109157S113405Bao, W., Jin, S., & Markowich, P. A. (2002). On Time-Splitting Spectral Approximations for the Schrödinger Equation in the Semiclassical Regime. Journal of Computational Physics, 175(2), 487-524. doi:10.1006/jcph.2001.6956Balakrishnan, N., Kalyanaraman, C., & Sathyamurthy, N. (1997). Time-dependent quantum mechanical approach to reactive scattering and related processes. Physics Reports, 280(2), 79-144. doi:10.1016/s0370-1573(96)00025-7Descombes, S., & Thalhammer, M. (2010). An exact local error representation of exponential operator splitting methods for evolutionary problems and applications to linear Schrödinger equations in the semi-classical regime. BIT Numerical Mathematics, 50(4), 729-749. doi:10.1007/s10543-010-0282-4Bader, P., Iserles, A., Kropielnicka, K., & Singh, P. (2014). Effective Approximation for the Semiclassical Schrödinger Equation. Foundations of Computational Mathematics, 14(4), 689-720. doi:10.1007/s10208-013-9182-8Gradinaru, V., & Hagedorn, G. A. (2013). Convergence of a semiclassical wavepacket based time-splitting for the Schrödinger equation. Numerische Mathematik, 126(1), 53-73. doi:10.1007/s00211-013-0560-6Keller, J., & Lasser, C. (2013). Propagation of Quantum Expectations with Husimi Functions. SIAM Journal on Applied Mathematics, 73(4), 1557-1581. doi:10.1137/120889186Gradinaru, V., Hagedorn, G. A., & Joye, A. (2010). Tunneling dynamics and spawning with adaptive semiclassical wave packets. The Journal of Chemical Physics, 132(18), 184108. doi:10.1063/1.3429607Gradinaru, V., Hagedorn, G. A., & Joye, A. (2010). Exponentially accurate semiclassical tunneling wavefunctions in one dimension. Journal of Physics A: Mathematical and Theoretical, 43(47), 474026. doi:10.1088/1751-8113/43/47/474026Coronado, E. A., Batista, V. S., & Miller, W. H. (2000). Nonadiabatic photodissociation dynamics ofICNin the à continuum: A semiclassical initial value representation study. The Journal of Chemical Physics, 112(13), 5566-5575. doi:10.1063/1.481130Church, M. S., Hele, T. J. H., Ezra, G. S., & Ananth, N. (2018). Nonadiabatic semiclassical dynamics in the mixed quantum-classical initial value representation. The Journal of Chemical Physics, 148(10), 102326. doi:10.1063/1.5005557Hagedorn, G. A. (1998). Raising and Lowering Operators for Semiclassical Wave Packets. Annals of Physics, 269(1), 77-104. doi:10.1006/aphy.1998.5843Faou, E., Gradinaru, V., & Lubich, C. (2009). Computing Semiclassical Quantum Dynamics with Hagedorn Wavepackets. SIAM Journal on Scientific Computing, 31(4), 3027-3041. doi:10.1137/080729724McLachlan, R. I. (1995). Composition methods in the presence of small parameters. BIT Numerical Mathematics, 35(2), 258-268. doi:10.1007/bf01737165Blanes, S., Casas, F., & Ros, J. (1999). Symplectic Integration with Processing: A General Study. SIAM Journal on Scientific Computing, 21(2), 711-727. doi:10.1137/s1064827598332497Blanes, S., Casas, F., & Ros, J. (2000). Celestial Mechanics and Dynamical Astronomy, 77(1), 17-36. doi:10.1023/a:1008311025472Blanes, S., Diele, F., Marangi, C., & Ragni, S. (2010). Splitting and composition methods for explicit time dependence in separable dynamical systems. Journal of Computational and Applied Mathematics, 235(3), 646-659. doi:10.1016/j.cam.2010.06.018Stefanov, B., Iordanov, O., & Zarkova, L. (1982). Interaction potential in1Σg+Hg2: fit to the experimental data. Journal of Physics B: Atomic and Molecular Physics, 15(2), 239-247. doi:10.1088/0022-3700/15/2/01

Exponentially Accurate Semiclassical Tunneling Wave Functions in One Dimension

We study the time behavior of wave functions involved in tunneling through a smooth potential barrier in one dimension in the semiclassical limit. We determine the leading order component of the wave function that tunnels. It is exponentially small in $1/\hbar$. For a wide variety of incoming wave packets, the leading order tunneling component is Gaussian for sufficiently small $\hbar$. We prove this for both the large time asymptotics and for moderately large values of the time variable

Convergence of a semiclassical wavepacket based time-splitting for the Schrödinger equation

We propose a new algorithm for solving the semiclassical time-dependent Schrödinger equation. The algorithm is based on semiclassical wavepackets. The focus of the analysis is only on the time discretization: convergence is proved to be quadratic in the time step and linear in the semiclassical parameter $$\varepsilon$$ ε

Whitney Elements on Sparse Grids

Da sich bei einigen Betriebssystemen die PDF-Datei nicht öffnen ließ, wurde eine weitere - technisch überarbeitete PDF-Datei - ergänzt.This work generalize the idea of the discretizations on sparse grids to differential forms. The extension to general l-forms in d dimensions includes the well known Whitney elements, as well as H(div)- and H(curl)- conforming mixed finite elements. The construction is based on one-dimensional differential forms, related wavelet representations and their tensor products. In addition to the construction of spaces, interpolation estimates are given. They display the typical efficiency of approximations based on sparse grids. Discrete inf-sup conditions are shown theoreticaly and experimentaly for mixed second order problems. The focus is on the stability of the discretization of the primal and of the dual mixed problem by sparse grid Whitney forms. The explanation of the involved algorithms received a particular attention, filling a gap in the literature. Details on the multilevel transforms, approximate interpolation operators, mass and stiffness matrix multiplications are given. The construction of general stencils on anisotropic full grids completes the detailed description of the multigrid solver based on semicoarsening

Acyl-CoA Dehydrogenasen: Mechanistische Untersuchungen mit der "Medium chain"-Acyl-CoA-Dehydrogenase

Acyl-CoA dehydrogenases constitute a family of flavoproteins that catalyze the a,b-dehydrogenation of fatty acid acyl-CoA thioesters. Medium chain acyl-CoA dehydrogenase (MCAD) is one of the best-studied members of this family. The a,b-dehydrogenation reaction involves the concerted C-H bonds cleavage of the substrate. First, the active site base, Glu376-COO-, removes a proton by and then a hydride is transferred to the flavin N(5) position of FAD. In my thesis MCAD several mechanistic details of the dehydrogenation reaction for MCAD were investigated. For this, among other things, a mutant of MCAD was created, which carries a C-terminal "His Tag". Addition of affinity His Tag facilitates purification of recombinant MCAD. For the investigation of the mechanism above several E376- or/and E99-MCAD mutants were used. Last one received an earlier attention since the Glu99 is located underneath of the active site of MCAD. This residue affects ionizations inside the active center cavity. Many studies were focused on E376Q-MCAD mutant. This mutant was highly inactive, because the glutamine does not play the role of the base. However its residual activity is 1/100000 of that of wtMCAD. This is a small value, but has the same order of magnitude as those found in non-catalyzed reactions. Proton inventory technique was suitable for mechanistic study of this mutant. Apart from this, it was observed that the log of rates of dehydrogenation increases linearly with the pH suggesting HO- as a reactant. A similar dependence was observed with Glu376Gln+Glu99Gly-MCAD. Thus, activity and reduction studies exclude Glu99 as a candidate for proton abstraction in the first step of dehydrogenation. E376Q-MCAD mutant reflected a large unexpected solvent isotope effect of approx. 8.5. The large isotope effects resulted from proton inventory experiments are attributed to the change in state of several H-bonds that occur during the process. A further investigation concerns the role of a special H-bond between N(5) of the flavin cofactor and Thr168-OH. However, an amino acid functional group that forms such a H-bond is strictly conserved in the ACAD familily (Thr or Ser). In the absence of this H-bond (T168A-MCAD) two effects could be observed: a) electronic influence on the substrate activation as well as on the redox potential of the flavin; b) steric - this H-bond is involved in the fine-tuning of the orientation of the flavin cofactor and ligand. Another threonine residue (Thr136) modulates the redox potential of the flavin (approx. -30 mV compared to wtMCAD 1.4 Kcal M-1). Thus e.g. with the Thr136Ala mutant the cofactor was partially reduced by the substrate, which is attributed to decrease of the redox potential. These experiments were supported by theoretical calculations, which were accomplished by Olga Dmitrenko working at Univ. of Delaware (USA) in Prof. R. Bach group

Semiclassical Dynamics in Several Spaces Dimensions with Wavepackets: New Ideas and Challenges

We present a technique for solving the time-dependent Schroedinger equation for small hbar in two or more space dimensions. The main idea is to expand the solution as a linear combination of semiclassical wave packets, but we use a new technique to avoid an unstable part of the computations.Non UBCUnreviewedAuthor affiliation: ETH ZurichPostdoctora