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Journal of Physics A Mathematical and General

Evasive levels in quantisation through wavepacket coupling: a semi-classical investigation

Pierre Amiot, BERTRAND G. GIRAUD

Published Aug 1, 1984• 0 citations
Publisher

Abstract

A new method is presented to introduce classical mechanics elements into the problem of obtaining the spectrum of an operator H(p,q). A finite-rank functional space is created by centring complex wavepackets on a discrete number of points on an equi-energy of the classical H(p,q) and by placing real wavepackets in the classically forbidden region. The latter span the active subspace, P, and the former the inactive subspace, Q, for an application of the method of Bloch-Horowitz. Depending on the rank retained for Q, some levels can be calculated with an extreme accuracy in a simple model while others remain evasive. A semi-classical study of the Green function in the inactive subspace Q, classically allowed, gives a clear explanation of this phenomenon and sheds new light on the significance of the semi-classical approximation for the propagator. An extension to the problem of barrier penetration is proposed.

DOI
10.1088/0305-4470/17/11/014

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A new method is presented to introduce classical mechanics elements into the problem of obtaining the spectrum of an operator H(p,q). A finite-rank functional space is created by centring complex wavepackets on a discrete number of points on an equi-energy of the classical H(p,q) and by placing real wavepackets in the classically forbidden region. The latter span the active subspace, P, and the former the inactive subspace, Q, for an application of the method of Bloch-Horowitz. Depending on the rank retained for Q, some levels can be calculated with an extreme accuracy in a simple model while others remain evasive. A semi-classical study of the Green function in the inactive subspace Q, classically allowed, gives a clear explanation of this phenomenon and sheds new light on the significance of the semi-classical approximation for the propagator. An extension to the problem of barrier penetration is proposed.
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Semiclassical approximations for tunneling processes usually involve complex trajectories or complex times. In this paper we use a previously derived approximation involving only real trajectories propagating in real time to describe the scattering of a Gaussian wavepacket by a finite square potential barrier. We show that the approximation describes both tunneling and interferences very accurately in the limit of small Plank's constant. We use these results to estimate the tunneling time of the wavepacket and find that, for high energies, the barrier slows down the wavepacket but that it speeds it up at energies comparable to the barrier height.
A simple and elegant semiclassical theory for propagating wavepackets is presented. The development of the theory avoids the explicit use of the Green-function formalism. It involves expanding the potential to a quadratic time-dependent form and the use of multiple reference trajectories. The classical analogue in phase space predicts the accuracy of the results. The theory is applied to the calculation of the autocorrelation function in a Coulomb field in one dimension. Remarkable agreement with an exact quantum calculation is obtained in many circumstances in spite of the Coulomb singularity. All the detailed quantum behaviour, including the long-time recurrences and the spectrum, is reproduced.
Among the many perplexing results of quantum mechanics is one that contradicts a result from introductory physics: the possibility of finding a quantum particle in a region that would be forbidden classically by energy conservation. An especially interesting example of this phenomenon with practical applications is quantum tunneling. Here we investigate the reasons for this puzzling result by focusing on the difference between how quantities like kinetic and potential energy are represented mathematically in classical and quantum mechanics. In quantum mechanics, physical observables, like energy, are represented by operators rather than real numbers. The consequences of this difference will be illustrated explicitly using a toy model in which the kinetic and potential energy operators are represented by 2×22 \times 22×2 matrices, which do not commute like their classical analogs. This model will then illustrate how classically perplexing results, like a quantum particle being found in the forbidden region, can arise.
One generic difficulty of most state-specific many-body formalisms using the Jeziorski-Monkhorst ansatz: ψ = Σ(μ)exp(T(μ))|φ(μ)>c(μ) for the wave-operators is the large number of redundant cluster amplitudes. The number of cluster amplitudes up to a given rank is many more in number compared to the dimension of the Hilbert Space spanned by the virtual functions of up to the same rank of excitations. At the same time, all inactive excitations--though linearly independent--are far too numerous. It is well known from the success of the contracted multi-reference configuration interaction (MRCI(SD)) that, at least for the inactive double excitations, their model space dependence (μ-dependence) is weak. Considerable simplifications can thus be obtained by using a partially internally contracted description, which uses the physically appealing approximation of taking the inactive excitations T(i) to be independent of the model space labels (μ-independent). We propose and implement in this paper such a formalism with internal contractions for inactive excitations (ICI) within Mukherjee's state-specific multi-reference coupled cluster theory (SS-MRCC) framework (referred to from now on as the ICI-SS-MRCC). To the extent the μ-independence of T(i) is valid, we expect the ICI-SS-MRCC to retain the conceptual advantages of size-extensivity yet using a drastically reduced number of cluster amplitudes without sacrificing accuracy. Moreover, greater coupling is achieved between the virtual functions reached by inactive excitations as a result of the internal contraction while retaining the original coupling term for the μ-dependent excitations akin to the parent theory. Another major advantage of the ICI-SS-MRCC, unlike the other analogous internally contracted theories, such as IC-MRCISD, CASPT2, or MRMP2, is that it can use relaxed coefficients for the model functions. However, at the same time it employs projection manifolds for the virtuals obtained from inactive n hole-n particle (nh-np) excitations on the entire reference function containing relaxed model space coefficients. The performance of the method has been assessed by applying it to compute the potential energy surfaces of the prototypical H(4); to the torsional potential energy barrier for the cis-trans isomerism in C(2)H(4) as well as that of N(2)H(2), automerization of cyclobutadiene, single point energy calculation of CH(2), SiH(2), and comparing them against the SS-MRCC results, benchmark full CI results, wherever available and those from the allied MR formalisms. Our findings are very much reminiscent of the experience gained from the IC-MRCISD method.