Oral: Non-commutative Optimal-Transport Informed Reduced Density MatrixFunctional Theory
ORAL
Abstract
In electronic structure theory there is typically a dichotomy between very accurate or exact methods on the one hand and very approximate methods on the other. They serve different purposes.
While highly accurate methods can generally describe small systems and specific phenomena well, their computational cost prohibits the description of larger systems. Conversely, large systems can be described by methods that are not very accurate. Prediction of phenomena that require high accuracy and are attributed to larger systems falls to extrapolation or conjecture.
First-order Reduced Density Matrix Theory (1RDMFT) addresses these problems, but with an exact theoretical framework. In contrast to methods like Hartree Fock and Kohn Sham DFT, which rely on a 1 Slater-determinant model wavefunction, the method describes the breaking of bonds qualitatively correct. At the same time, it does not fall into the category of Quantum Merlin Arthur complexity class such as FCI or the N-representability problem in Second-order reduced Density Matrix Theory (2RDMFT), which are methods that can describe highly correlated systems such as a dissociating bond.
First-order Reduced Density Matrix Theory approximates the diagonal of the second order reduced density matrix (2RDM) to predict the electron-electron interaction without having to find the actual wavefunction. Previous attempts at 1RDMFT have been made by Goedecker and Umrigar, Buijese and Baerends, Mueller, all in a very similar way, by trying to find approximate 2RDM terms as functionals of the natural orbitals and occupation numbers by comparing with exact results in model systems [1-3].
In our novel strategy, we use regularized quantum optimal transport theory to provide a framework for 1RDMFT. This gives us access to previously unexplored relationships between fundamental properties, such as scaling properties or definitive bounds. By accessing these, we develop new 1RDMFT methods that we believe will fill the gap in electronic structure theory and allow predictions that could previously only be inferred.
[1] A.M.K. Müller. Physics Letters A 105.9 pp.446–452., (1984).
[2] O. Gritsenko, K. Pernal, E. Baerends The Journal of Chemical Physics 122, p. 204102, (2005).
[3] S. Goedecker and C. J. Umrigar. Physical Review Letters 81, pp. 866–869, (1998).
While highly accurate methods can generally describe small systems and specific phenomena well, their computational cost prohibits the description of larger systems. Conversely, large systems can be described by methods that are not very accurate. Prediction of phenomena that require high accuracy and are attributed to larger systems falls to extrapolation or conjecture.
First-order Reduced Density Matrix Theory (1RDMFT) addresses these problems, but with an exact theoretical framework. In contrast to methods like Hartree Fock and Kohn Sham DFT, which rely on a 1 Slater-determinant model wavefunction, the method describes the breaking of bonds qualitatively correct. At the same time, it does not fall into the category of Quantum Merlin Arthur complexity class such as FCI or the N-representability problem in Second-order reduced Density Matrix Theory (2RDMFT), which are methods that can describe highly correlated systems such as a dissociating bond.
First-order Reduced Density Matrix Theory approximates the diagonal of the second order reduced density matrix (2RDM) to predict the electron-electron interaction without having to find the actual wavefunction. Previous attempts at 1RDMFT have been made by Goedecker and Umrigar, Buijese and Baerends, Mueller, all in a very similar way, by trying to find approximate 2RDM terms as functionals of the natural orbitals and occupation numbers by comparing with exact results in model systems [1-3].
In our novel strategy, we use regularized quantum optimal transport theory to provide a framework for 1RDMFT. This gives us access to previously unexplored relationships between fundamental properties, such as scaling properties or definitive bounds. By accessing these, we develop new 1RDMFT methods that we believe will fill the gap in electronic structure theory and allow predictions that could previously only be inferred.
[1] A.M.K. Müller. Physics Letters A 105.9 pp.446–452., (1984).
[2] O. Gritsenko, K. Pernal, E. Baerends The Journal of Chemical Physics 122, p. 204102, (2005).
[3] S. Goedecker and C. J. Umrigar. Physical Review Letters 81, pp. 866–869, (1998).
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Publication: Implementation of the Exact Functional in 1-electron Reduced Density Matrix Theory: An Optimal Transport Approach
Presenters
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Jannis T Erhard
McMaster University
Authors
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Jannis T Erhard
McMaster University
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Paul Ayers
McMaster University