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High dimensional quantum dynamics: challenges and opportunities |
KEYNOTE LECTURES Quantum approaches for
dynamics of high dimensional systems Joel M. Bowman,^{a} Bastiaan Braams,^{a,b}
Stuart Carter,^{a,c} Xinchuan Huang,^{a} Zhong Jin^{a}, Zhen Xie^{a} ^{ } ^{ } 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 10^{4}-10^{5}
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. CH_{5}^{+}, H_{5}^{+}, H_{5}O_{2}^{+},
H_{3}O_{2}^{-}, CH_{3}OH, 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, Heidelberg, Germany 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 H_{2}+H_{2} 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 Uwe Manthe 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+CH_{4}_{ }→H_{2}+CH_{3} 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 Peter Saalfrank 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 G.A. Worth 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) [3],
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 [8]. 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 [9]
and passage through a conical intersection [10].
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 [10]. 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. [1] H.-D. Meyer,
U. Manthe and L. S. Cederbaum, Chem. Phys. Lett. 165 (1990) 73. [2] M. H. Beck,
A. Jäckle, G. A. Worth and H.-D. Meyer, Phys. Rep. 324 (2000) 1. [3] G. A.
Worth, J. Chem. Phys. 112 (2000) 8322. [4] H.-D. Meyer
and G. A. Worth, Theor. Chem. Acc. 109 (2003) 251. [5] H. Wang
and M. Thoss, J. Chem. Phys. 119 (2003) 1289. [6]
I. Burghardt, H.-D. Meyer and L. S. Cederbaum, J. Chem. Phys.
111 (1999) 2927. [7]
I. Burghardt, M. Nest and G. A. Worth, J. Chem. Phys. 119 (2003)
5364. [8] H. Wang,
M. Thoss and W. H. Miller, J. Chem. Phys. 115 (2001) 2979. [9] G. Worth and I. Burghardt, Chem. Phys.
Lett. 368 (2003) 502. [10] G. A. Worth, M. A. Robb and
I. Burghardt, Farad. Discuss. 127 (2004) 307. INVITED LECTURES
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 braams@mathcs.emory.edu 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
[1]). 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. [1]
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 Ove Christiansen Department of Chemistry, University
of Århus, Denmark, ove@chem.au.dk Recently
a new formulation for the description of the dynamics of molecular systems has
been developed[1]. 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)[5], 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. [4] 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). 4. O. Christiansen, J. Chem. Phys. 122,194105
(2005). 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) [1] 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 H_{2}+H
initial state selected scattering calculations as example. Beck et al. [2] 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 [3]. 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 [4]. 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 H_{2}+Cu(100) reactive scattering as
examples. [1]
U. Manthe, J. Chem. Phys. 105, 6989 (1996). [2]
M. H. Beck et al., Physics Reports 324, 1 (2000) [3]
R. van Harrevelt and U. Manthe, J. Chem. Phys. 121, 5623 (2004). [4]
R. Dawes and T. Carrington Jr., J.
Chem. Phys. 121, 726 (2004). Reduced density matrix descriptions of gas-surface scattering at finite temperature Bret Jackson Department of Chemistry, University of MassachusettsAmherst, 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 Horst Köppel 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 [1]. A brief outline
is given of our specific quantum dynamical approach how to deal with these
problems and phenomena [2], and a more recent extension presented [3]. 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 [4] 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. [1] Conical intersections.
Electronic structure, dynamics and spectroscopy, W. Domcke, D.R. Yarkony and H.
Köppel (Eds.), World Scientific (Singapore) 2004. [2] H. Köppel, W. Domcke and L.S. Cederbaum, in [1],
p. 323-367. [3] H. Köppel, Faraday Discuss.
127 (2004) 35-47 [4] E.V. Gromov et al., J.
Chem. Phys. 121 (2004)
4585-4598. [5] M. Döscher, H. Köppel
and P.G. Szalay, J. Chem. Phys. 117
(2004) 2645-2656. [6] H. Köppel et al., J. Chem.
Phys. 117 (2002) 2657-2671. Multidimensional
quantum dynamics of intramolecular hydrogen bonds
Oliver Kühn Institut für Chemie,
Physikalische und Theoretische Chemie, Freie Universität Berlin, Takustr. 3, D-14195 Berlin, ok@chemie.fu-berlin.de 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. [1]
B. A. Ruf, W. H. Miller, J. Chem. Soc. Faraday Trans. 2, 84, 1523, (1988). [2]
K. Giese, O. Kühn, J. Chem. Phys., in press. [3] H. Naundorf, G. A. Worth, H.-D.
Meyer, O. Kühn, J. Phys. Chem. A 106, 719 (2002). [4]
M. Petković, O. Kühn, J. Phys. Chem. A 107, 8458 (2003). Correlated
Many Electron Dynamics with theMulti-Configuration Time-Dependent Hartree-Fock
(MCTDHF) method Mathias Nest Universität
Potsdam 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 [Back] |