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High dimensional quantum dynamics: challenges and opportunities
Quantum approaches for dynamics of high dimensional systems
Joel M. Bowman,a Bastiaan Braams,a,b Stuart Carter,a,c
Xinchuan Huang,a Zhong Jina, Zhen Xiea
I will begin my talk with a review of some of the bottlenecks in doing quantum dynamics calculations for larger molecules, including the potential energy surface. I will then describe some strategies that have been incorporated into the code MULTIMODE to overcome some of these bottlenecks. Advances in creating potential energy surfaces that explicitly exploit the permutational symmetry of like atoms will also be reviewed briefly. (A much more detailed presentation of the methods and algorithms developed for this purpose will be given in the talk by Braams.) This approach is used to fit of the order of 104-105 high quality ab initio energies to obtain global potentials that also dissociate correctly.
I will next describe applications in full dimensionality to a variety of molecular systems, eg. CH5+, H5+, H5O2+, H3O2-, CH3OH, etc. Some of these applications will include diffusion Monte Carlo calculations done for the zero-point energy by us and for excited states done by Anne McCoy (Ohio State).
a Department of Chemistry and Cherry L. Emerson Center for Scientific Computation
b Department of Mathematics and Computer Science and Cherry L. Emerson Center for Scientific Computation, Emory University, Atlanta, GA
c Department of Chemistry, University of Reading, UK
We thank the U.S. Department of Energy, the U.S. National Science Foundation and the U.S. Office of Naval Research for financial support.
Michael A. Collins
Research School of Chemistry, Australian National University,
Canberra. ACT 0200 Australia
Molecular potential energy surfaces (PESs) which describe chemical reactions can now be constructed from ab initio calculations in an automated way, for molecules containing up to several atoms. The method, established and refined over the last decade, relies on interpolation of calculated ab initio data over the whole of the relevant space of molecular configurations. Recently, progress has been achieved on a similar approach to constructing the diabatic potential energy matrix that describes reactions which involve more than one electronic state. To make substantial progress in understanding more complex reaction mechanisms, we must devise means for constructing chemically accurate energy surfaces for substantially larger molecules. This talk will present recent progress towards this goal, and will (hopefully) discuss how the evaluation of PES can be made friendlier for MCTDH calculations.
Applications of MCTDH to Intramolecular vibrational-energy redistribution and scatering problems of tetra- and penta-atomic molecular systems
Fabien Gatti (a)
Hans-Dieter Meyer (b)
(a) Laboratoire LSDSMS, UMR 5636-CNRS, CC0014
Université Montpellier II,
34095 Montpellier, Frans
(b) Theoretische Chemie, Universität Heidelberg,
Im Neuenheimer Feld 229, D-69120,
Accurate quantum calculations on the pectroscopy and dynamic for highly deformable molecular systems are a challenging task in environmental science, astrophysics and chemical reactivity. We present several applications of the Multi-configuration Time Dependent Hartree code from Heidelberg (1) which demonstrate that MCTDH is an efficient tool to investigate rotational excitation in H2+H2 collisions or the IVR (Intramolecular vibrational-energy redistribution) in relatively large systems such as HONO, HFCO (6 degrees of freedom), Toluene and Fluoroform (9 degrees of freedom). In order to study these systems the use of curvilinear coordinates is essential to describe motions of large amplitude but unfortunately, generally leads to very involved expressions of the kinetic energy operator. We therefore present a novel and general approach to describe molecules based on a polyspherical description of molecular systems (2). This approach explicitelly provides general and simple expressions of the exact kinetic energy operators whatever the number of atoms and whatever the vector parametrization (for instance Jacobi or valence vectors) and in a form perfectly adapted to MCTDH. Moreover, for a full efficiency of MCTDH, the potential energy surfaces must be provided in a particular form, i.e. a sum of products of operators which act on a 'limited' number of degrees of freedom. We show how the recently generalized potfit scheme (3,4) allows us to fit the 6- or 9-dimensional PESs in this adapted form.
(1) H.-D. Beyer, U. Manthe, L.S. Cederbaum Chem.Phys. Letters 165 (1990) 73 and G.A. Worth, M.H. Beck, A. Jäckle, and H.-D. Meyer, the MCTDH package, version 8.2 (2000). H.-D. Meyer, version 8.3 (2002) See http://www.pci.uni-heidelberg.de/tc/usr/mctdh.
(2) F. Gatti, C. Iung, M. Menou, Y. Justum, A. Nauts, X. Chapuisat. J. Chem. Phys. 108 (1998) 8821.
(3) A. Jäckle and H.-D. Meyer J. Chem. Phys. 109 (1998) 3772.
(4) F. Gatti, H.-D. Meyer Chem. Phys. 304 (2004) 3.
Quantendynamik polyatomarer Reaktionen
Theoretische Chemie, Fakultät für Chemie, Universität Bielefeld
Universitätsstr. 25, 33615 Bielefeld, Germany
Quantum effect play an important role in the dynamics of many chemical reactions. Tunneling strongly enhances the reactions rates of hydrogen transfer reactions. Ultrafast non-adiabatic transitions due to conical intersections of electronic potential energy surfaces frequently occur in photochemical reactions. The accurate theoretical description of these quantum effects is a challenging task for polyatomic systems: the numerical effort of quantum dynamics calculations typically increases exponentially with the number of relevant degrees of freedom and the construction of multi-dimensional potential energy surfaces faces similar problems. The multi-configurational time-dependent Hartree (MCTDH) approach offers the unique possibility for accurate calculations of polyatomic reactions beyond the range of other exact quantum dynamics methods. Combined with the flux correlation function approach for “direct” reaction rate calculations, the MCTDH method is particularly efficient for the calculation of thermal rate constants and cumulative reaction probabilities. Results of MCTDH calculations for different thermal and photoinduced reactions will be presented. These will include recent accurate reaction rate calculations for the H+CH4 →H2+CH3 reaction and its isotopic variants employing an accurate potential energy surface obtained by Shepard interpolation, calculations on the dynamics of Ethen following π 2 →π π * excitation, and studies of dissociative absorption processes on surfaces.
System-bath methods and applications
Institut für Chemie, Universität Potsdam, D-14476 Potsdam-Golm
In this contribution time-resolved quantum dynamical "system-bath" approaches, their performance, and their application to selected elementary processes in condensed phases will be presented.
We first give a general overview over system-bath methods to treat complex multi-mode problems, with special emphasis on open-system density matrix theory. Various simple examples for few-level systems will illustrate concepts such as energy and phase relaxation.
To treat molecular systems in an environment more realistically, efficient numerical methods are required, some of which will be described in this lecture.
Finally, various applications of the theory, mostly in the field of surface science will be described. Where possible, we compare "exact" methods based on the dynamics of the full system, to "reduced" dynamics. Also, connections to MCTDH as applied to reduced and full dynamics will be made at various places.
New Directions in MCTDH: Towards Truly Polyatomic Molecules and General Photochemistry
Dept. of Chemistry, University of Birmingham, Edgbaston,
Birmingham, B15 2TT, U.K.
Many systems of interest in photochemistry have a large number of degrees of freedom. To study the dynamics after photoexcitation it is also often necessary to use a full quantum dynamical treatment. This is especially the case if non-adiabatic effects are important, e.g a conical intersection is present along the pathway. The MCTDH method [1, 2] is known for its power and ability to treat systems using full quantum dynamics larger than those accessible to standard wavepacket methods. Even so, in its basic form we are restricted to fewer than 30 modes. In order to treat molecules larger than benzene without resorting to reduced dimensionality models we thus need to develop new approaches.
In this talk various approaches will be covered that attempt this. One branch is the efficient implementation of full MCTDH, either by selection of important configurations (S-MCTDH) , or by using MCTDH to propagate the single-particle functions, i.e. the “cascading” or “multi-layered” method [4, 5]. The other, independent, direction is to use parametrised functions for some of the single-particle functions, e.g. Gaussian functions (G-MCTDH) [6, 7], or even replacing spfs with classical trajectories as done in the work of Thoss and Wang .
The G-MCTDH method leads naturally to a variational method for propagating Gaussian wavepackets (vMCG). In contrast to most other Gaussian wavepacket methods, the basis functions are coupled directly, and do not follow classical trajectories. As a result the method converges much quicker and are able to directly describe quantum events such as tunneling  and passage through a conical intersection . As the vMCG method uses local basis functions, it opens the door to a quantum dynamics method suitable for “direct dynamics” in which the potential energy surface is calculated on-the-fly by a quantum chemistry program . This not only promises to remove the tedium of fitting multi-dimensional potential surfaces, but it should provide a model-free way of gaining information on a process.
 H.-D. Meyer, U. Manthe and L. S. Cederbaum, Chem. Phys. Lett. 165 (1990) 73.
 M. H. Beck, A. Jäckle, G. A. Worth and H.-D. Meyer, Phys. Rep. 324 (2000) 1.
 G. A. Worth, J. Chem. Phys. 112 (2000) 8322.
 H.-D. Meyer and G. A. Worth, Theor. Chem. Acc. 109 (2003) 251.
 H. Wang and M. Thoss, J. Chem. Phys. 119 (2003) 1289.
 I. Burghardt, H.-D. Meyer and L. S. Cederbaum, J. Chem. Phys. 111 (1999) 2927.
 I. Burghardt, M. Nest and G. A. Worth, J. Chem. Phys. 119 (2003) 5364.
 H. Wang, M. Thoss and W. H. Miller, J. Chem. Phys. 115 (2001) 2979.
 G. Worth and I. Burghardt, Chem. Phys. Lett. 368 (2003) 502.
 G. A. Worth, M. A. Robb and I. Burghardt, Farad. Discuss. 127 (2004) 307.
Full-Dimensional, Permutationally Invariant Potential Energy and
Dipole Moment Surfaces: Mathematical and Computational Aspects
Bastiaan J. Braams
Department of Mathematics and Computer Science
and Emerson Center for Scientific Computation
Emory University - Atlanta, GA 30322
Over the past two years we have constructed full-dimensional potential energy surfaces for a variety of molecular systems, among them H5+, CH5, CH5+, H3O2-, H4O2, H5O2+, C2H2O, C3H3O, CH2O, C3H2, and HOONO; for some of these systems we also fitted a dipole moment surface. (A sample reference is ). The property of invariance under permutations of like nuclei is built into the basis for the least-squares fitting procedure. The use of a cluster expansion (many-body expansion), going up to five-body or at most six-body terms, caters for dissociation and reaction processes and also for extension to larger systems. The fitted potential and its gradient are evaluated on a millisecond timescale, making it possible to do molecular dynamics or quantum Monte Carlo calculations at ab initio accuracy without anywhere near the cost that is normally associated with ab initio MD, or even with a Car-Parrinello treatment. The fitted surfaces have also been used for MULTIMODE calculations of a vibrational spectrum.
Some studies of reaction dynamics and spectroscopy that relied on these surfaces will be described by J. M. Bowman at this workshop. The present talk will focus on the mathematical and computational aspects. I will describe how the MAGMA computer algebra system was used to help create Fortran code for the evaluation of a generating family of permutationally invariant polynomials, and so of a basis for the least squares fitting. I will also discuss the concept of covariants and their use for fitting vector or tensor quantities, notably the dipole moment. Some practical matters to be described include the use of gradient and hessian information, and the iterative process of fitting, testing the fit, and generating suitable sample configurations for further ab initio calculations to improve the surface.
The codes used for generating and evaluating the fitted surfaces have already been used outside the Emory environment and can be made more widely available; some necessary polishing is under way this summer. The codes include a core mathematical library to evaluate the invariant polynomials for a wide variety of molecular symmetry groups, and a library on top of that to construct and evaluate the surfaces.
 X. Huang, B. J. Braams, and J. M. Bowman: "Ab initio potential energy and dipole moment surfaces for H5O2+". Journal of Chemical Physics 122 (Jan 22, 2005) 044308.
Choosing 1-D basis functions to facilitate extracting a good multidimensional basis from a huge direct product basis:
coupled systems with as many as 16 coordinates
Tucker Carrington Jr
Département de chimie, Université de Montréal
In this talk I shall discuss a new scheme for choosing basis functions for quantum dynamics calculations. Direct product bases are frequently used to do quantum dynamics calculations. The number of direct product functions required to converge a spectrum, compute a rate constant etc. is very large. Using the MCTDH scheme enables one to reduce the number of product basis functions required to describe short term dynamics but it is difficult to compute very accurate vibrational or ro-vibrational spectra. If one employs a time-independent approach there are two obvious ways to decrease the number of basis functions required to obtain converged energy levels. One can either use contracted basis functions that are eigenfunctions of reduced-dimension Hamiltonians incorporating some of the coupling or discard unnecessary functions from an initial huge direct product basis. We have used both approaches but in this talk I shall discuss new ways of choosing 1-d functions from which to build product basis sets and demonstrate that if the 1-d functions are chosen wisely a very large number can be discarded. The 1-D functions from which we build the direct product basis are chosen to satisfy two conditions: (1) they nearly diagonalise the full Hamiltonian matrix; (2) they minimize off-diagonal matrix elements that couple basis functions with diagonal elements close to those of the energy levels we wish to compute. By imposing these conditions we increase the number of product functions that can be removed from the multidimensional basis without degrading the accuracy of computed energy levels. Two basic types of 1-D basis functions are in common use for time-independent calculations: eigenfunctions of 1-D Hamiltonians and discrete variable representation (DVR) functions. Both have advantages and disadvantages. The 1-D functions we propose are intermediate between the 1-D eigenfunction functions and the DVR functions. If the coupling is very weak, they are very nearly 1-D eigenfunction functions. As the strength of the coupling is increased they resemble more closely DVR functions. I shall talk about using trace minimization to define parameters that define the 1-D basis functions. Results will be presented for a family of exactly solvable model problems with as many as 16 coordinates.
New methods for calculating vibrational wave functions, energies and molecular properties: Vibrational coupled cluster theory
Department of Chemistry, University of Århus,
Recently a new formulation for the description of the dynamics of molecular systems has been developed. The formalism is similar in spirit to the second quantization formulation of electronic structure theory. On the basis of this formalism a new program package and a number of new computational methods and new implementations of existing methods have been developed[2-4] and are currently being further developed for applications in many different contexts. The primary focus here is on methods for calculating bound states, in particular vibrational wave functions.
In this talk we shall consider the theory and use of vibrations self consistent field (VSCF), vibrational Møller Plesseth perturbation theory (VMP)[6,3], vibrational configuration interaction (VCI)[7,2] and vibrational coupled cluster (VCC)[1,2]. I shall describe benchmark calculation relating to various convergence issues with respect to order and/or truncations in the parameter space. In this context the concept of size-extensivity is also discussed. It is shown that the new VCC approach gives higher accuracy compared to more standard approaches with the same number of parameters. On the hand the VCC method gives rise to new problems in the implementation.
I will describe certain aspects of the implementation in the MidasCpp program aiming at allowing VSCF, VMP2, VCI, and VCC calculations on large molecular systems. Examples of calculations on systems with many degrees of freedom will be described. Finally, new initiatives towards using the developed methods in calculating vibrational contributions to molecular properties shall be discussed. 
1. O. Christiansen, J.Chem.Phys, 120, 2140 (2004).
2. O. Christiansen, J.Chem.Phys, 120, 2149 (2004).
3. O. Christiansen, J.Chem.Phys, 119, 5773 (2003).
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5. J.M.Bowman, J.Chem.Phys, 68, 608 (1978).
6. L.S. Norris, M.A. Ratner, A.E. Roitberg, and R.B. Gerber, J.Chem.Phys, 105, 11261 (1996).
7. S. Carter, J.M. Bowman, and N.C. Handy, Theor. Chim. Acta. 100, 191 (1998).
The correlation discrete variable representation
Rob van Harrevelt
Technische Universitaet Muenchen, Lichtenbergstrasse 4,
85747 Garching, Germany.
The correlation discrete variable representation (CDVR)  can be used to evaluate matrix elements of potentials in MCTDH calculations. This approach, which is based on a time-dependent discrete variable representation (DVR), does not pose a restriction on the expression of the potential energy operator. This talk discusses recent developments of the CDVR method.
First, a symmetry effect in the CDVR approach is discussed, using H2+H initial state selected scattering calculations as example. Beck et al.  have reported a breakdown of the CDVR scheme for this system. This breakdown is caused by symmetry present in the initial state selected scattering calculation. Employing a symmetry-adapted CDVR scheme, the problem can be solved and accurate results can be obtained .
Second, the extension of the CDVR method to MCTDH calculations employing mode combination is discussed. Mode combination implies that some degrees of freedom are grouped together and treated as a single mode in the MCTDH approach. This can significantly reduce the computational effort in high-dimensional MCTDH calculations. However, the CDVR approach for mode combination calculations requires a general multidimensional non direct product DVR, because the single-particle functions for the combined modes are multidimensional functions. Such a multidimensional DVR has recently been proposed by Dawes and Carrington . In this talk an implementation of their scheme in MCTDH/CDVR calculations is discussed. The accuracy of the multidimensional CDVR approach is studied using photodissociation of NOCl and H2+Cu(100) reactive scattering as examples.
 U. Manthe, J. Chem. Phys. 105, 6989 (1996).
 M. H. Beck et al., Physics Reports 324, 1 (2000)
 R. van Harrevelt and U. Manthe, J. Chem. Phys. 121, 5623 (2004).
 R. Dawes and T. Carrington Jr., J. Chem. Phys. 121, 726 (2004).
Reduced density matrix descriptions of gas-surface scattering at finite temperature
Department of Chemistry, University of Massachusetts
Amherst, MA 01003, USA
A formalism is developed for modeling the interaction between a particle scattering from a surface and the thermal vibrations of the lattice, using the reduced density matrix. A short time propagator is constructed for the reduced density matrix, with the dissipative terms derived from the full particle-bath Hamiltonian. The equations obey detailed balance and accurately describe the relaxation to equilibrium at long times. The method is used to study the scattering, trapping, sticking and desorption of atoms from corrugated non-rigid surfaces. The effects of thermal motion on diffraction mediated selective adsorption (trapping) are examined, as are the effects of corrugation on sticking.
Multi-state multi-mode vibronic dynamics in spectroscopy and photoinduced processes
Theoretische Chemie, Universität Heidelberg, Im Neuenheimer Feld 229,
D-69120 Heidelberg, Germany
Nuclear dynamics on conically intersecting potential energy surfaces is known to be strongly nonadiabatic and to involve an intricate mixing between electronic and nuclear degrees of freedom as well as coupling between the various vibrational modes themselves . A brief outline is given of our specific quantum dynamical approach how to deal with these problems and phenomena , and a more recent extension presented . The pertinent coupling constants are extracted from high-level ab initio calculations. Applications to treat larger systems (often with more than two strongly coupled potential energy surfaces) have relied, in particular, on the MCTDH method. They comprise singlet excited states of furan  and the lowest five doublet electronic states of the benzene radical cation [5,6]. Good agreement with experimental spectral profiles is achieved. The quantum dynamical calculations display a rich variety of nonadiabatic phenomena, in the energy and time domain. Some of these will be presented in relation to the above examples, and their importance to the photodynamics, also in general, will be pointed out.
 Conical intersections. Electronic structure, dynamics and spectroscopy, W. Domcke, D.R. Yarkony and H. Köppel (Eds.), World Scientific (Singapore) 2004.
 H. Köppel, W. Domcke and L.S. Cederbaum, in , p. 323-367.
 H. Köppel, Faraday Discuss. 127 (2004) 35-47
 E.V. Gromov et al., J. Chem. Phys. 121 (2004) 4585-4598.
 M. Döscher, H. Köppel and P.G. Szalay, J. Chem. Phys. 117 (2004) 2645-2656.
 H. Köppel et al., J. Chem. Phys. 117 (2002) 2657-2671.
Multidimensional quantum dynamics of intramolecular hydrogen bonds
Institut für Chemie, Physikalische und Theoretische Chemie, Freie Universität Berlin,
Takustr. 3, D-14195 Berlin, email@example.com
The reaction path motion of a Hydrogen atom across an intramolecular Hydrogen bond is often characterized by a simultaneous rearrangement of the molecule’s heavy atoms. Thus, a rigorous theoretical description faces the challenges of generating a multidimensional potential energy surface and solving the respective Schrödinger equation for the nuclear wave function.
As a compromise between accuracy and feasibility reaction surface concepts have been established. Here an accurate treatment of the large amplitude reactive motion of the Hydrogen atom is combined with a harmonic approximation for the skeleton degrees of freedom. An all Cartesian formulation [1,2] was shown to be particularly convenient because it contains all couplings between the coordinates in the potential energy operator. Besides having an approximate but full-dimensional Hamiltonian this approach has the additional advantage that the latter has a form which is separable in most degrees of freedom. This facilitates an effective numerical solution of the time-dependent Schrödinger equation by using the Multiconfiguration Time-Dependent Hartree method [3,4].
In this contribution a number of examples for multidimensional nuclear quantum dynamics of Hydrogen bonds will be discussed. Emphasis will be put on the interpretation of complex spectral lineshapes in terms of time dependent wave packets and related ultrafast infrared spectroscopies.
 B. A. Ruf, W. H. Miller, J. Chem. Soc. Faraday Trans. 2, 84, 1523, (1988).
 K. Giese, O. Kühn, J. Chem. Phys., in press.
 H. Naundorf, G. A. Worth, H.-D. Meyer, O. Kühn, J. Phys. Chem. A 106, 719 (2002).
 M. Petković, O. Kühn, J. Phys. Chem. A 107, 8458 (2003).
Correlated Many Electron Dynamics with theMulti-Configuration Time-Dependent Hartree-Fock (MCTDHF) method
We present the Multi-Configuration Time-Dependent Hartree-Fock (MCTDHF) method, which is an adaptation of the MCTDH method to the correlated quantum dynamics of systems of many electrons. The correlated motion of electrons plays an essential role in many branches of physics and chemistry, like photoelectron spectroscopy, scanning tunneling microscopy, etc. This is even more so, since laser pulses on the atto-second time scale now allow to study the time-resolved dynamics of electron systems out of equilibrium.
The MCTDHF method is applied to various problems, like excitation/ionization by laser pulses, inelastic electron scattering, inverse photoemission, and electron impact ionization. We discuss the properties of the method in the framework of one-dimensional model systems. Familiar concepts from nuclear quantum dynamics, namely Complex Absorbing Potentials, flux analysis, propagation in imaginary time and the wave packet approach to spectroscopy are applied to correlated many electron dynamics, and the differences and new aspects are discussed. Some comparisons with another method for many electron dynamics, the time-dependent configuration interaction singles (TD-CIS) method, are made.
M. Nest, T. Klamroth, P. Saalfrank, J. Chem. Phys. 122, 124102 (2005)
M. Nest, T. Klamroth, Phys. Rev. A, accepted