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CRC Press eBooks

Introduction

Hiroki Nakamura, Gennady Mil'nikov

Published Jul 18, 2013• 1 citation
Publisher

Abstract

There are enormous amounts of literature dealing with quantum mechanical tunneling phenomena, andmanymonographs have also been published recently covering various fields of the phenomenon [21-24]. Unfortunately, however, theories of multidimensional tunneling phenomena are not yet satisfactorily developed. Effective numerical algorithms in quantum mechanics such as MCTDH (multichannel timedependent Hartree) method [25] and quantum Monte Carlo method [26] have been developed and successfully implemented to study quantum dynamics in multidimensional systems. However, exact quantum dynamics methods require a formidable task including accurate ab initio calculations of global potential energy surfaces. Besides, it is not straightforward to obtain the physical insights of dynamic processes.

DOI
10.1201/b14673-5

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In the case of tunneling in chemical reactions [see figure1.1(c)], the total energy of reagents cannot be generally considered to be small; treating it as the first-order term of h¯ as was successfully done in Chapters 6 and 7 does not lead to any sensible semiclassical theory. The semiclassical mechanics requires (1) construction of Lagrange manifolds, (2) determination of amultidimensional caustic surface, and (3) connection of wave functions between classically allowed and forbidden regions [59]. Unfortunately, these problems have not yet been solved mathematically. As was discussed in Section 3.1, in the framework of the ordinary WKB approach we can construct Lagrange manifolds and caustics in the case of two dimensions and the connection problem can be treated relatively well by assuming the local separability. Extension of this method to higher dimensions, however, cannot be practical. The complexvalued (complex coordinate and/or momentum) WKB theory has been devised, as briefly explained there, but cannot be simply extended to multidimensions (see, for instance, [15]). This formidable task can hardly be accomplished mainly because of the complexity of the Lagrange manifold and the fact that analytical properties of any realistic potential energy surfaces extended into the complex-valued coordinate space cannot be guaranteed. The use of a complex trajectory was nicely materialized by the double-ended complex trajectory in the classical S-matrix theory [206], in which the quantized boundary conditions are satisfied at both (initial and final) ends. Unfortunately, this method turned out not to be practically useful, because finding such trajectories is also a formidable task in a multidimensional complex coordinate (and/or momentum) space. We can, however, make a break through the difficulty of detecting caustics in multidimensional space. An efficient way of doing that has been devised along classical trajectory [47]. This is not a trivial task at all, since it is not possible to find caustics in high-dimensional space by just shooting classical trajectories. Since the caustics can be found along each classical trajectory, chemical dynamics with tunneling effects incorporated can now be treated by the on-the-fly method. In this subsection, this method is explained.

CRC Press eBooks•Jul 18, 2013•2 citations
There are enormous amounts of literature dealing with quantum mechanical tunneling phenomena, andmanymonographs have also been published recently covering various fields of the phenomenon [21-24]. Unfortunately, however, theories of multidimensional tunneling phenomena are not yet satisfactorily developed. Effective numerical algorithms in quantum mechanics such as MCTDH (multichannel timedependent Hartree) method [25] and quantum Monte Carlo method [26] have been developed and successfully implemented to study quantum dynamics in multidimensional systems. However, exact quantum dynamics methods require a formidable task including accurate ab initio calculations of global potential energy surfaces. Besides, it is not straightforward to obtain the physical insights of dynamic processes.
The coupled coherent states method has demonstrated itself as an accurate and efficient method of studying the quantum dynamics of various systems. In recent years, its applicability has been extended by incorporating a number of new numerical expansions and modifications to generate a closely related family of methods. In this thesis, two new augmentations are developed to further broaden the scope of problems that are able to be treated. The first of these is a 2-layer extension of coupled coherent states, capable of providing an increased mathematical description of a degree or degrees of freedom within a quantum mechanical system, as well as beneficial numerical and scalability properties. The newly developed method is tested on a model system-bath Hamiltonian consisting of a tunnelling mode governed by an asymmetric double well potential coupled to a harmonic bath. It is found to compare well to previous methods of studying the Hamiltonian, as well as a benchmark calculation on the system conducted in this thesis, and demonstrate the beneficial numerical and scalability properties expected. The second development is to extend coupled coherent states to treat systems of indistinguishable bosons in the second quantisation representation. The method is tested on the same Hamiltonian as the 2-layer coupled coherent states scheme, where the harmonic bath is second quantised as it is comprised of oscillators of the same frequency, so they may be thought of as indistinguishable. Exploiting this symmetry property is found to be extremely advantageous, with remarkable agreement to the benchmark calculation. The method is then tested on a model Hamiltonian consisting of 100 bosons in a shifted harmonic trap, with oscillations in the 1-body density calculated. The results are found to compare favourably with a multiconfigurational time-dependent Hartree for bosons calculation that is equivalent to the Gross-Pitaevskii equation, providing impetus for future studies on systems of Bose-Einstein condensates. The existing ab initio multiple cloning extension of coupled coherent states for nonadiabatic dynamics is also used to study the ultrafast photodissociation of 2-ethylpyrrole. The results are compared to experimental data, and a novel insight into the dissociation mechanism is obtained, with it shown to be composed of a two step process. Firstly, molecules that are able to dissociate immediately over the barrier along the N-H coordinate do so in < 50 fs, and this is followed by a second slower dissociation process from molecules that must sample the potential energy surface before finding a way around the barrier. This is not observed experimentally due to the temporal widths of the laser pulses obscuring the dynamics in the < 50 fs window.
While key effects of the many-body problem---such as Kondo and Mott physics---can be understood in terms of on-site correlations, non-local fluctuations of charge, spin, and pairing amplitudes are at the heart of the most fascinating and unresolved phenomena in condensed matter physics. Here, we review recent progress in diagrammatic extensions to dynamical mean-field theory for ab initio materials calculations. We first recapitulate the quantum field theoretical background behind the two-particle vertex. Next we discuss latest algorithmic advances in quantum Monte Carlo simulations for calculating such two-particle quantities using worm sampling and vertex asymptotics, before giving an introduction to the ab initio dynamical vertex approximation (AbinitioDΓΓΓA). Finally, we highlight the potential of AbinitioDΓΓΓA by detailing results for the prototypical correlated metal SrVO3_33​.
In the case of tunneling in chemical reactions [see figure1.1(c)], the total energy of reagents cannot be generally considered to be small; treating it as the first-order term of h¯ as was successfully done in Chapters 6 and 7 does not lead to any sensible semiclassical theory. The semiclassical mechanics requires (1) construction of Lagrange manifolds, (2) determination of amultidimensional caustic surface, and (3) connection of wave functions between classically allowed and forbidden regions [59]. Unfortunately, these problems have not yet been solved mathematically. As was discussed in Section 3.1, in the framework of the ordinary WKB approach we can construct Lagrange manifolds and caustics in the case of two dimensions and the connection problem can be treated relatively well by assuming the local separability. Extension of this method to higher dimensions, however, cannot be practical. The complexvalued (complex coordinate and/or momentum) WKB theory has been devised, as briefly explained there, but cannot be simply extended to multidimensions (see, for instance, [15]). This formidable task can hardly be accomplished mainly because of the complexity of the Lagrange manifold and the fact that analytical properties of any realistic potential energy surfaces extended into the complex-valued coordinate space cannot be guaranteed. The use of a complex trajectory was nicely materialized by the double-ended complex trajectory in the classical S-matrix theory [206], in which the quantized boundary conditions are satisfied at both (initial and final) ends. Unfortunately, this method turned out not to be practically useful, because finding such trajectories is also a formidable task in a multidimensional complex coordinate (and/or momentum) space. We can, however, make a break through the difficulty of detecting caustics in multidimensional space. An efficient way of doing that has been devised along classical trajectory [47]. This is not a trivial task at all, since it is not possible to find caustics in high-dimensional space by just shooting classical trajectories. Since the caustics can be found along each classical trajectory, chemical dynamics with tunneling effects incorporated can now be treated by the on-the-fly method. In this subsection, this method is explained.