Symplectic integration and physical interpretation of time-dependent coupled-cluster theory

The formulation of the time-dependent Schrodinger equation in terms of coupled-cluster theory is outlined, with emphasis on the bivariational framework and its classical Hamiltonian structure. An indefinite inner product is introduced, inducing physical interpretation of coupled-cluster states in the form of transition probabilities, autocorrelation functions, and explicitly real values for observables, solving interpretation issues which are present in time-dependent coupled-cluster theory and in ground-state calculations of molecular systems under influence of external magnetic fields. The problem of the numerical integration of the equations of motion is considered, and a critial evaluation of the standard fourth-order Runge--Kutta scheme and the symplectic Gauss integrator of variable order is given, including several illustrative numerical experiments. While the Gauss integrator is stable even for laser pulses well above the perturbation limit, our experiments indicate that a system-dependent upper limit exists for the external field strengths. Above this limit, time-dependent coupled-cluster calculations become very challenging numerically, even in the full configuration interaction limit. The source of these numerical instabilities is shown to be rapid increases of the amplitudes as ultrashort high-intensity laser pulses pump the system out of the ground state into states that are virtually orthogonal to the static Hartree-Fock reference determinant.Comment: 14 pages, 13 figure

Fermion NN-representability for prescribed density and paramagnetic current density

The $N$-representability problem is the problem of determining whether or not there exists $N$-particle states with some prescribed property. Here we report an affirmative solution to the fermion $N$-representability problem when both the density and paramagnetic current density are prescribed. This problem arises in current-density functional theory and is a generalization of the well-studied corresponding problem (only the density prescribed) in density functional theory. Given any density and paramagnetic current density satisfying a minimal regularity condition (essentially that a von Weiz\"acker-like the canonical kinetic energy density is locally integrable), we prove that there exist a corresponding $N$-particle state. We prove this by constructing an explicit one-particle reduced density matrix in the form of a position-space kernel, i.e.\ a function of two continuous position variables. In order to make minimal assumptions, we also address mathematical subtleties regarding the diagonal of, and how to rigorously extract paramagnetic current densities from, one-particle reduced density matrices in kernel form

Ab initio quantum dynamics using coupled-cluster

The curse of dimensionality (COD) limits the current state-of-the-art {\it ab initio} propagation methods for non-relativistic quantum mechanics to relatively few particles. For stationary structure calculations, the coupled-cluster (CC) method overcomes the COD in the sense that the method scales polynomially with the number of particles while still being size-consistent and extensive. We generalize the CC method to the time domain while allowing the single-particle functions to vary in an adaptive fashion as well, thereby creating a highly flexible, polynomially scaling approximation to the time-dependent Schr\"odinger equation. The method inherits size-consistency and extensivity from the CC method. The method is dubbed orbital-adaptive time-dependent coupled-cluster (OATDCC), and is a hierarchy of approximations to the now standard multi-configurational time-dependent Hartree method for fermions. A numerical experiment is also given.Comment: 5 figure

A state-specific multireference coupled-cluster method based on the bivariational principle

A state-specific multireference coupled-cluster method based on Arponen's bivariational principle is presented, the bivar-MRCC method. The method is based on singlereference theory, and therefore has a relatively straightforward formulation and modest computational complexity. The main difference from established methods is the bivariational formulation, in which independent parameterizations of the wavefunction (ket) and its complex conjugate (bra) are made. Importantly, this allows manifest multiplicative separability (exact in the extended bivar-MRECC version of the method, and approximate otherwise), while preserving polynomial scaling of the working equations. A feature of the bivariational principle is that the formal bra and ket references can be included as bivariational parameters, which eliminates much of the bias towards the formal reference. A pilot implementation is described, and extensive benchmark calculations on several standard problems are performed. The results from the bivar-MRCC method are comparable to established state-specific multireference methods. Considering the relative affordability of the bivar-MRCC method, it may become a practical tool for non-experts

Geometry of effective Hamiltonians

We give a complete geometrical description of the effective Hamiltonians common in nuclear shell model calculations. By recasting the theory in a manifestly geometric form, we reinterpret and clarify several points. Some of these results are hitherto unknown or unpublished. In particular, commuting observables and symmetries are discussed in detail. Simple and explicit proofs are given, and numerical algorithms are proposed, that improve and stabilize common methods used today.Comment: 1 figur

Computing all Pairs (λ,μ) Such That λ is a Double Eigenvalue of Α + μΒ

Double eigenvalues are not generic for matrices without any particular structure. A matrix depending linearly on a scalar parameter, Α + μΒ, will, however, generically have double eigenvalues for some values of the parameter μ. In this paper, we consider the problem of finding those values. More precisely, we construct a method to accurately find all scalar pairs (λ,μ) such that Α + μΒ has a double eigenvalue λ, where Α and Β are given arbitrary complex matrices. The general idea of the globally convergent method is that if μ is close to a solution, then Α + μΒ has two eigenvalues which are close to each other. We fix the relative distance between these two eigenvalues and construct a method to solve and study it by observing that the resulting problem can be stated as a two-parameter eigenvalue problem, which is already studied in the literature. The method, which we call the method of fixed relative distance (MFRD), involves solving a two-parameter eigenvalue problem which returns approximations of all solutions. It is unfortunately not possible to get full accuracy with MFRD. In order to compute solutions with full accuracy, we present an iterative method which returns a very accurate solution, for a sufficiently good starting value. The approach is illustrated with one academic example and one application to a simple problem in computational quantum mechanics. Copyright 2011 Society for Industrial and Applied Mathematic