Personalia
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Research interests
- Ab initio quantum mechanics: theory and numerics:
- Coupled Cluster theory
- Multiconfigurational time-dependent Hartree method (MCTDH) and its variants
- Non-linear Schrödinger equations, e.g., the Gross-Pitaevskii equation
- Perturbation theory
- Numerical linear algebra
Publications
| Halvorsen, T. & Kvaal, S., "Manifestly gauge invariant discretizations of the Schrödinger equation", Accepted for publication in Physics Letters A 376, 1107-1114 (2012) |
| Abstract: Grid-based discretizations of the time dependent Schrödinger equation coupled to an external magnetic field are converted to manifest gauge invariant discretizations. This is done using generalizations of ideas used in classical lattice gauge theory, and the process defined is applicable to a large class of discretized differential operators. In particular, popular discretizations such as pseudospectral discretizations using the fast Fourier transform can be transformed to gauge invariant schemes. Also generic gauge invariant versions of generic time integration methods are considered, enabling completely gauge invariant calculations of the time dependent Schrödinger equation. Numerical examples illuminating the differences between a gauge invariant discretization and conventional discretization procedures are also presented. |
| BibTeX: @article{Halvorsen2012, author = {Halvorsen, T.G. and Kvaal, S.}, title = {Manifestly gauge invariant discretizations of the Schrödinger equation}, journal = {Accepted for publication in Physics Letters A}, year = {2012}, volume = {376}, pages = {1107--1114}, url = {http://www.sciencedirect.com/science/article/pii/S0375960112001636} } |
| Hjorth-Jensen, M., Dean, D., Hagen, G. & Kvaal, S., "Many-body interactions and nuclear structure", J. Phys. G: Nucl. Part. Phys. 37, 064035 (2010) |
| Abstract: This paper presents several challenges to nuclear many-body theory and our understanding of the stability of nuclear matter. In order to achieve this, we present five different cases, starting with an idealized toy model. These cases expose problems that need to be understood in order to match recent advances in nuclear theory with current experimental programs in low-energy nuclear physics. In particular, we focus on our current understanding, or lack thereof, of many-body forces, and how they evolve as functions of the number of particles. We provide examples of discrepancies between theory and experiment and outline some selected perspectives for future research directions. |
| BibTeX: @article{HjorthJensen2010, author = {Hjorth-Jensen, M. and Dean, D.J. and Hagen, G. and Kvaal, S.}, title = {Many-body interactions and nuclear structure}, journal = {J. Phys. G: Nucl. Part. Phys.}, year = {2010}, volume = {37}, number = {6}, pages = {064035}, url = {http://stacks.iop.org/0954-3899/37/i=6/a=064035} } |
| Jarlebring, E., Kvaal, S. & Michiels, W., "Computing all Pairs (lambda, mu) Such That lambda is a Double Eigenvalue of A + mu B", SIAM Journal on Matrix Analysis and Applications 32, 902-927 (2011) |
| Abstract: Double eigenvalues are not generic for matrices without any particular structure. A matrix depending linearly on a scalar parameter, A+μB, 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 A+μB has a double eigenvalue λ, where A and B are given arbitrary complex matrices. The general idea of the globally convergent method is that if μ is close to a solution, then A+μB 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. |
| BibTeX: @article{Jarlebring2011, author = {Jarlebring, E. and Kvaal, S. and Michiels, W.}, title = {Computing all Pairs (lambda, mu) Such That lambda is a Double Eigenvalue of A + mu B}, journal = {SIAM Journal on Matrix Analysis and Applications}, publisher = {SIAM}, year = {2011}, volume = {32}, number = {3}, pages = {902-927}, url = {http://link.aip.org/link/?SML/32/902/1}, doi = {http://dx.doi.org/10.1137/100783157} } |
| Kvaal, S., "Ab initio quantum dynamics using coupled-cluster", The Journal of Chemical Physics 136, 194109 (2012) |
| Abstract: The curse of dimensionality (COD) limits the current state-of-the-art 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 Schrödinger equation. The method inherits size-consistency and extensivity from the CC method. The method is dubbed orbital-adaptive time-dependent coupled-cluster, and is a hierarchy of approximations to the now standard multi-configurational time-dependent Hartree method for fermions. A numerical experiment is also given. |
| BibTeX: @article{Kvaal2012, author = {Simen Kvaal}, title = {Ab initio quantum dynamics using coupled-cluster}, journal = {The Journal of Chemical Physics}, publisher = {AIP}, year = {2012}, volume = {136}, number = {19}, pages = {194109}, url = {http://link.aip.org/link/?JCP/136/194109/1}, doi = {http://dx.doi.org/10.1063/1.4718427} } |
| Kvaal, S., "Multiconfigurational time-dependent Hartree method for describing particle loss due to absorbing boundary conditions", Phys. Rev. A 84, 022512 (2011) |
| Abstract: Absorbing boundary conditions in the form of a complex absorbing potential are routinely introduced in the Schrödinger equation to limit the computational domain or to study reactive scattering events using the multi-configurational time-dependent Hartree method (MCTDH). However, it is known that a pure wave-function description does not allow the modeling and propagation of the remnants of a system of which some parts are removed by the absorbing boundary. It was recently shown [S. Selstø and S. Kvaal, J. Phys. B: At. Mol. Opt. Phys. bfseries 43 (2010), 065004] that a master equation of Lindblad form was necessary for such a description. We formulate a multiconfigurational time-dependent Hartree method for this master equation, usable for any quantum system composed of any mixture of species. The formulation is a strict generalization of pure-state propagation using standard MCTDH. We demonstrate the formulation with a numerical experiment. |
| BibTeX: @article{Kvaal2011a, author = {Kvaal, S.}, title = {Multiconfigurational time-dependent Hartree method for describing particle loss due to absorbing boundary conditions}, journal = {Phys. Rev. A}, year = {2011}, volume = {84}, pages = {022512}, url = {http://pra.aps.org/abstract/PRA/v84/i2/e022512} } |
| Kvaal, S., "Harmonic oscillator eigenfunction expansions, quantum dots, and effective interactions", Phys. Rev. B 80, 045321 (2009) |
| Abstract: We give a thorough analysis of the convergence properties of the configuration-interaction method as applied to parabolic quantum dots among other systems, including a priori error estimates. The method converges slowly in general, and in order to overcome this, we propose to use an effective two-body interaction well known from nuclear physics. Through numerical experiments we demonstrate a significant increase in accuracy of the configuration-interaction method. |
| BibTeX: @article{Kvaal2009, author = {Kvaal, Simen }, title = {Harmonic oscillator eigenfunction expansions, quantum dots, and effective interactions}, journal = {Phys. Rev. B}, publisher = {American Physical Society}, year = {2009}, volume = {80}, number = {4}, pages = {045321}, doi = {http://dx.doi.org/10.1103/PhysRevB.80.045321} } |
| Kvaal, S., "Open source FCI code for quantum dots and effective interactions" (2008) |
| Abstract: We describe OpenFCI, an open source implementation of the full configuration-interaction method (FCI) for two-dimensional quantum dots with optional use of effective renormalized interactions. The code is written in C++ and is available under the Gnu General Public License. The code and core libraries are well documented and structured in a way such that customizations and generalizations to other systems and numerical methods are easy tasks. As examples we provide a matrix element tabulation program and an implementation of a simple model from nuclear physics, in addition to the quantum dot application itself. |
| BibTeX: @unpublished{Kvaal2008, author = {Kvaal, S.}, title = {Open source FCI code for quantum dots and effective interactions}, year = {2008}, note = {arXiv:0810.2644v1}, url = {http://arxiv.org/abs/0810.2644v1} } |
| Kvaal, S., "Geometry of effective Hamiltonians", Phys. Rev. C 78, 044330 (2008) |
| Abstract: 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 methods commonly used today. |
| BibTeX: @article{Kvaal2008a, author = {Kvaal, S.}, title = {Geometry of effective Hamiltonians}, journal = {Phys. Rev. C}, publisher = {APS}, year = {2008}, volume = {78}, number = {4}, pages = {044330}, url = {http://link.aps.org/abstract/PRC/v78/e044330}, doi = {http://dx.doi.org/10.1103/PhysRevC.78.044330} } |
| Kvaal, S., Hjorth-Jensen, M. & Møll Nilsen, H., "Effective interactions, large-scale diagonalization, and one-dimensional quantum dots", Phys. Rev. B 76, 085421 (2007) |
| Abstract: The widely used large-scale diagonalization method using harmonic oscillator basis functions (an instance of the Rayleigh-Ritz method [S. Gould, Variational Methods for Eigenvalue Problems: An Introduction to the Methods of Rayleigh, Ritz, Weinstein, and Aronszajn (Dover, New York, 1995)], also called a spectral method, configuration-interaction method, or "exact diagonalization" method) is systematically analyzed using results for the convergence of Hermite function series. We apply this theory to a Hamiltonian for a one-dimensional model of a quantum dot. The method is shown to converge slowly, and the nonsmooth character of the interaction potential is identified as the main problem with the chosen basis, while, on the other hand, its important advantages are pointed out. An effective interaction obtained by a similarity transformation is proposed for improving the convergence of the diagonalization scheme, and numerical experiments are performed to demonstrate the improvement. Generalizations to more particles and dimensions are discussed. |
| BibTeX: @article{Kvaal2007, author = {Kvaal, S. and Hjorth-Jensen, M. and Møll Nilsen, H.}, title = {Effective interactions, large-scale diagonalization, and one-dimensional quantum dots}, journal = {Phys. Rev. B}, year = {2007}, volume = {76}, pages = {085421}, doi = {http://dx.doi.org/10.1103/PhysRevB.76.085421} } |
| Kvaal, S., Jarlebring, E. & Michiels, W., "Computing singularities of perturbation series", Phys. Rev. A 83, 032505 (2011) |
| Abstract: Many properties of current ab initio approaches to the quantum many-body problem, both perturbational and otherwise, are related to the singularity structure of the Rayleigh-Schrödinger perturbation series. A numerical procedure is presented that in principle computes the complete set of singularities, including the dominant singularity which limits the radius of convergence. The method approximates the singularities as eigenvalues of a certain generalized eigenvalue equation which is solved using iterative techniques. It relies on computation of the action of the Hamiltonian matrix on a vector and does not rely on the terms in the perturbation series. The method can be useful for studying perturbation series of typical systems of moderate size, for fundamental development of resummation schemes, and for understanding the structure of singularities for typical systems. Some illustrative model problems are studied, including a helium-like model with δ-function interactions for which Møller-Plesset perturbation theory is considered and the radius of convergence found. |
| BibTeX: @article{Kvaal2011, author = {Kvaal, S. and Jarlebring, E. and Michiels, W.}, title = {Computing singularities of perturbation series}, journal = {Phys. Rev. A}, year = {2011}, volume = {83}, pages = {032505}, doi = {http://dx.doi.org/10.1103/PhysRevA.83.032505} } |
| Pedersen Lohne, M., Hagen, G., Hjorth-Jensen, M., Kvaal, S. & Pederiva, F., "Ab initio computation of the energies of circular quantum dots", Phys. Rev. B 84, 115302 (2011) |
| Abstract: We perform coupled-cluster and diffusion Monte Carlo calculations of the energies of circular quantum dots up to 20 electrons. The coupled-cluster calculations include triples corrections and a renormalized Coulomb interaction defined for a given number of low-lying oscillator shells. Using such a renormalized Coulomb interaction brings the coupled-cluster calculations with triples correlations in excellent agreement with the diffusion Monte Carlo calculations. This opens up perspectives for doing ab initio calculations for much larger systems of electrons. |
| BibTeX: @article{PedersenLohne2011, author = {Pedersen Lohne, M. and Hagen, G. and Hjorth-Jensen, M. and Kvaal, S. and Pederiva, F.}, title = {Ab initio computation of the energies of circular quantum dots}, journal = {Phys. Rev. B}, year = {2011}, volume = {84}, pages = {115302}, url = {http://prb.aps.org/abstract/PRB/v84/i11/e115302}, doi = {http://dx.doi.org/10.1103/PhysRevB.84.115302} } |
| Selstø, S., Birkeland, T., Kvaal, S., Nepstad, R. & Førre, M., "The role of final state correlation in double ionization of helium: a master equation approach", J. Phys. B: At. Mol. Opt. Phys. 44, 215003 (2011) |
| Abstract: It is demonstrated how a numerical approach based on absorbing boundaries may be used to describe the process of non-sequential two-photon double ionization of helium. Contrary to any method based on solving the Schrödinger equation alone, this numerical scheme is able to reconstruct the remaining particles as one particle is absorbed. This may be used to distinguish between single and double ionization. A model of reduced dimensionality, which describes the process at a qualitative level, has been used. The results have been compared with a more conventional method in which the time-dependent Schrödinger equation is solved and the final wavefunction is analysed in terms of projection onto eigenstates of the uncorrelated Hamiltonian, i.e. with no electron–electron interaction included in the final states. It is found that the two methods indeed produce the same total cross sections for the process. |
| BibTeX: @article{Selsto2011, author = {Selstø, S. and Birkeland, T. and Kvaal, S. and Nepstad, R. and Førre, M.}, title = {The role of final state correlation in double ionization of helium: a master equation approach}, journal = {J. Phys. B: At. Mol. Opt. Phys.}, year = {2011}, volume = {44}, pages = {215003}, url = {http://iopscience.iop.org/0953-4075/44/21/215003/}, doi = {http://dx.doi.org/10.1088/0953-4075/44/21/215003} } |
| Selstø, S. & Kvaal, S., "Absorbing boundary conditions for dynamical many-body quantum systems", J. Phys. B: At. Mol. Opt. Phys. 43, 065004 (2010) |
| Abstract: In numerical studies of the dynamics of unbound quantum mechanical systems, absorbing boundary conditions are frequently applied. Although this certainly provides a useful tool in facilitating the description of the system, its applications to systems consisting of more than one particle are problematic. This is due to the fact that all information about the system is lost upon the absorption of one particle; a formalism based solely on the Schrödinger equation is not able to describe the remainder of the system as particles are lost. Here we demonstrate how the dynamics of a quantum system with a given number of identical fermions may be described in a manner which allows for particle loss. A consistent formalism which incorporates the evolution of sub-systems with a reduced number of particles is constructed through the Lindblad equation. Specifically, the transition from an N -particle system to an ( N - 1)-particle system due to a complex absorbing potential is achieved by relating the Lindblad operators to annihilation operators. The method allows for a straight forward interpretation of how many constituent particles have left the system after interaction. We illustrate the formalism using one-dimensional two-particle model problems. |
| BibTeX: @article{Selsto2010, author = {Selstø, S. and Kvaal, S.}, title = {Absorbing boundary conditions for dynamical many-body quantum systems}, journal = {J. Phys. B: At. Mol. Opt. Phys.}, year = {2010}, volume = {43}, number = {6}, pages = {065004}, url = {http://stacks.iop.org/0953-4075/43/i=6/a=065004} } |
Colleagues (not yet complete)
- Lars Bojer Madsen
- Snorre H. Christiansen
- Tore G. Halvorsen
- Daniel J. Haxton
- Morten Hjorth-Jensen
- Sebastian Holtz
- Elias Jarlebring
- Christian Lubich
- Wim Michiels
- Xavier Raynaud
- Thorsten Rohwedder
- Reinhold Schneider
- Sølve Selstø
Misc
- OpenFCI v0.6 is now available. The code is currently work in progress.
- Introduction to approximation of derivatives using the discrete
Fourier transform and the FFT in Matlab, as a script
fft_intro.m - Computing abscissa and weights of Gauss-Legendre
quadrature
gauleg.m - The large, sparse and symmetric standard eigenvalue problem
using ARPACK in Matlab (there is a major bug in the usual eigs.m in
this case):
symeig.m - A simple two-electron quandum dot using pseudospectral
discretization on an interval
qdot2.m