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Variational determination of the two-particle density matrix: the case of doubly-occupied space

Ward Poelmans (UGent)
(2015)
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(UGent) and (UGent)
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Abstract
The world at the level of the atom is described by the branch of science called quantum mechanics. The crown jewel of quantum mechanics is given by the Schrödinger equation which describes a system of indistinguishable particles, that interact with each other. However, an equation alone is not enough: the solution is what interests us. Unfortunately, the exponential scaling of the Hilbert space makes it unfeasible to calculate the exact wave function. This dissertation concerns itself with one of the many ab initio methods that were developed to solve this problem: the variational determination of the second-order density matrix. This method already has a long history. It is not considered to be on par with best ab initio methods. This work tries an alternative approach. We assume that the wave function has a Slater determinant expansion where all orbitals are doubly occupied or empty. This assumption drastically reduces the scaling of the N-representability conditions. The downside is that the energy explicitly depends on the used orbitals and thus an orbital optimizer is needed. The hope is that by using this approximation, we can capture the lion's share of the static correlation and that any missing dynamic correlation can be added through perturbation theory. We developed an algorithm based on Jacobi rotations. The scaling is much more favorable compared to the general case. The method is then tested on a array of benchmark systems.
Keywords
N-representability, DOCI, v2DM

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Citation

Please use this url to cite or link to this publication:

Chicago
Poelmans, Ward. 2015. “Variational Determination of the Two-particle Density Matrix: The Case of Doubly-occupied Space”. Ghent, Belgium: Ghent University. Faculty of Sciences.
APA
Poelmans, W. (2015). Variational determination of the two-particle density matrix: the case of doubly-occupied space. Ghent University. Faculty of Sciences, Ghent, Belgium.
Vancouver
1.
Poelmans W. Variational determination of the two-particle density matrix: the case of doubly-occupied space. [Ghent, Belgium]: Ghent University. Faculty of Sciences; 2015.
MLA
Poelmans, Ward. “Variational Determination of the Two-particle Density Matrix: The Case of Doubly-occupied Space.” 2015 : n. pag. Print.
@phdthesis{6933577,
  abstract     = {The world at the level of the atom is described by the branch of science called quantum mechanics. The crown jewel of quantum mechanics is given by the Schr{\"o}dinger equation which describes a system of indistinguishable particles, that interact with each other. However, an equation alone is not enough: the solution is what interests us. Unfortunately, the exponential scaling of the Hilbert space makes it unfeasible to calculate the exact wave function.
This dissertation concerns itself with one of the many ab initio methods that were developed to solve this problem: the variational determination of the second-order density matrix. This method already has a long history. 
It is not considered to be on par with best ab initio methods.
This work tries an alternative approach. We assume that the wave function has a Slater determinant expansion where all orbitals are doubly occupied or empty. This assumption drastically reduces the scaling of the N-representability conditions. The downside is that the energy explicitly depends on the used orbitals and thus an orbital optimizer is needed. The hope is that by using this approximation, we can capture the lion's share of the static correlation and that any missing dynamic correlation can be added through perturbation theory.
We developed an algorithm based on Jacobi rotations. The scaling is much more favorable compared to the general case. The method is then tested on a array of benchmark systems.},
  author       = {Poelmans, Ward},
  isbn         = {9789461973030},
  language     = {eng},
  pages        = {XX, 225},
  publisher    = {Ghent University. Faculty of Sciences},
  school       = {Ghent University},
  title        = {Variational determination of the two-particle density matrix: the case of doubly-occupied space},
  year         = {2015},
}