Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1602.00578

Concise configuration interaction expansions for three fermions in six orbitals

Alex D. Gottlieb, Norbert J. Mauser, J. M. Zhang

Published Feb 1, 2016• 1 citation
arXivHTMLPDF

Abstract

The Hilbert space for three fermions in six orbitals, lately dubbed the "Borland-Dennis setting," is a proving ground for insights into electronic structure. Borland and Dennis discovered that, when referred to coordinate systems defined in terms of its natural orbitals, a wave function in the Borland-Dennis setting has the same structure as a 3-qubit state. By dint of the Borland-Dennis Theorem, canonical forms for 3-qubit states have analogs in the Borland-Dennis setting. One of these canonical forms is based upon "max-overlap Slater determinant approximations." Any max-overlap Slater determinant approximation of a given wave function is the leading term in a 5-term configuration interaction (CI) expansion of that wave function. Our main result is that "max-overlap CIS approximations" also lead to 5-term CI expansions, distinct from those based on max-overlap Slater determinant approximations, though of the same symmetric shape. We also prove the analog of this result for 3-qubit setting.

DOI
10.48550/arxiv.1602.00578

Related papers

Best Slater approximation of a fermionic wave function

Alex D. Gottlieb, Norbert J. Mauser, J. M. Zhang

We study the best Slater approximation of an $N$-fermion wave function analytically. That is, we seek the Slater determinant (constructed out of $N$ orthonormal single-particle orbitals) wave function having largest overlap with a given $N$-fermion wave function. Some simple lemmas have been established and their usefulness is demonstrated on some structured states, such as the GHZ state. In the simplest nontrivial case of three fermions in six orbitals, which the celebrated Borland-Dennis discovery is about, the best Slater approximation wave function is proven to be built out of the natural orbitals in an interesting way. We also show that the Hadamard inequality is useful for finding the best Slater approximation of some special target wave functions.

•Oct 19, 2015•0 citations

Optimal Slater-determinant approximation of fermionic wave functions

Jian‐Min Zhang, Norbert J. Mauser

We study the optimal Slater-determinant approximation of an $N$-fermion wave function analytically. That is, we seek the Slater-determinant (constructed out of $N$ orthonormal single-particle orbitals) wave function having largest overlap with a given $N$-fermion wave function. Some simple lemmas have been established and their usefulness is demonstrated on some structured states, such as the Greenberger-Horne-Zeilinger state. In the simplest nontrivial case of three fermions in six orbitals, which the celebrated Borland-Dennis discovery is about, the optimal Slater approximation wave function is proven to be built out of the natural orbitals in an interesting way. We also show that the Hadamard inequality is useful for finding the optimal Slater approximation of some special target wave functions.

arXiv:1510.05634•Physical Review A•Sep 23, 2016•12 citations

Structure of the first order reduced density matrix in three electron systems: A generalized Pauli constraints assisted study

Iris Theophilou, Nektarios N. Lathiotakis, Nicole Helbig

We investigate the structure of the one-body reduced density matrix of three electron systems, i.e., doublet and quadruplet spin configurations, corresponding to the smallest interacting system with an open-shell ground state. To this end, we use configuration interaction (CI) expansions of the exact wave function in Slater determinants built from natural orbitals in a finite dimensional Hilbert space. With the exception of maximally polarized systems, the natural orbitals of spin eigenstates are generally spin dependent, i.e., the spatial parts of the up and down natural orbitals form two different sets. A measure to quantify this spin dependence is introduced and it is shown that it varies by several orders of magnitude depending on the system. We also study the ordering issue of the spin-dependent occupation numbers which has practical implications in reduced density matrix functional theory minimization schemes, when generalized Pauli constraints (GPCs) are imposed and in the form of the CI expansion in terms of the natural orbitals. Finally, we discuss the aforementioned CI expansion when there are GPCs that are almost "pinned."

arXiv:1712.08670•The Journal of Chemical Physics•Mar 20, 2018•4 citations

Structure of the first order reduced density matrix in three electron systems: A generalized Pauli constraints assisted study

Theophilou, I., Lathiotakis, N., Helbig, N.

We investigate the structure of the one-body reduced density matrix of three electron systems, i.e., doublet and quadruplet spin configurations, corresponding to the smallest interacting system with an open-shell ground state. To this end, we use configuration interaction (CI) expansions of the exact wave function in Slater determinants built from natural orbitals in a finite dimensional Hilbert space. With the exception of maximally polarized systems, the natural orbitals of spin eigenstates are generally spin dependent, i.e., the spatial parts of the up and down natural orbitals form two different sets. A measure to quantify this spin dependence is introduced and it is shown that it varies by several orders of magnitude depending on the system. We also study the ordering issue of the spin-dependent occupation numbers which has practical implications in reduced density matrix functional theory minimization schemes, when generalized Pauli constraints (GPCs) are imposed and in the form of the CI expansion in terms of the natural orbitals. Finally, we discuss the aforementioned CI expansion when there are GPCs that are almost “pinned.”

•Mar 21, 2018•4 citations

Factorization of the wave-function coefficients in configuration-interaction expansions

Stefano Evangelisti

A factorization of the wave-function coefficients obtained from configuration-interaction expansions is presented. For this purpose, the system orbitals are partitioned into a set of overlapping domains and a factor is associated with a given electron distribution in each domain. The configuration-interaction coefficient of each Slater determinant is given by the product of all the domain factors that are associated with this determinant. This expansion is asymptotically exact, which means that the exact wave function is recovered in the limit of large domains. Moreover, the number of factors needed to define a wave function has a slow growth as a function of the system size. The idea is illustrated through applications to two model systems, i.e. a collection of non-interacting electron pairs and a Néel-type wave function. A possible strategy for the calculation of the configuration factors is also briefly discussed.

Molecular Physics•Mar 20, 2005•0 citations

Are natural orbitals optimal for full configuration interaction wave functions

Klaas J. H. Giesbertz

We investigate whether the natural orbitals (NOs) minimize $\lVert\Psi - \Phi\rVert^2$, where $\Psi$ is a wave function and $\Phi$ is a full configuration interaction (CI) approximation to $\Psi$ in a truncated basis. We will show that the NOs rarely provide the optimal orbitals for $\Phi$, except when (1) there are only two particles or (2) only one basis function is removed in the case of fermions. Further, we will show that the CI expansion coefficients of $\Psi$ and $\Phi$ are identical up to a global scaling factor and demonstrate how the NOs can be used to generate the orbitals that minimize $\lVert\Psi - \Phi\rVert^2$.

arXiv:1309.2449•Sep 10, 2013•0 citations
The Hilbert space for three fermions in six orbitals, lately dubbed the "Borland-Dennis setting," is a proving ground for insights into electronic structure. Borland and Dennis discovered that, when referred to coordinate systems defined in terms of its natural orbitals, a wave function in the Borland-Dennis setting has the same structure as a 3-qubit state. By dint of the Borland-Dennis Theorem, canonical forms for 3-qubit states have analogs in the Borland-Dennis setting. One of these canonical forms is based upon "max-overlap Slater determinant approximations." Any max-overlap Slater determinant approximation of a given wave function is the leading term in a 5-term configuration interaction (CI) expansion of that wave function. Our main result is that "max-overlap CIS approximations" also lead to 5-term CI expansions, distinct from those based on max-overlap Slater determinant approximations, though of the same symmetric shape. We also prove the analog of this result for 3-qubit setting.
We study the best Slater approximation of an NNN-fermion wave function analytically. That is, we seek the Slater determinant (constructed out of NNN orthonormal single-particle orbitals) wave function having largest overlap with a given NNN-fermion wave function. Some simple lemmas have been established and their usefulness is demonstrated on some structured states, such as the GHZ state. In the simplest nontrivial case of three fermions in six orbitals, which the celebrated Borland-Dennis discovery is about, the best Slater approximation wave function is proven to be built out of the natural orbitals in an interesting way. We also show that the Hadamard inequality is useful for finding the best Slater approximation of some special target wave functions.
We study the optimal Slater-determinant approximation of an NNN-fermion wave function analytically. That is, we seek the Slater-determinant (constructed out of NNN orthonormal single-particle orbitals) wave function having largest overlap with a given NNN-fermion wave function. Some simple lemmas have been established and their usefulness is demonstrated on some structured states, such as the Greenberger-Horne-Zeilinger state. In the simplest nontrivial case of three fermions in six orbitals, which the celebrated Borland-Dennis discovery is about, the optimal Slater approximation wave function is proven to be built out of the natural orbitals in an interesting way. We also show that the Hadamard inequality is useful for finding the optimal Slater approximation of some special target wave functions.
We investigate the structure of the one-body reduced density matrix of three electron systems, i.e., doublet and quadruplet spin configurations, corresponding to the smallest interacting system with an open-shell ground state. To this end, we use configuration interaction (CI) expansions of the exact wave function in Slater determinants built from natural orbitals in a finite dimensional Hilbert space. With the exception of maximally polarized systems, the natural orbitals of spin eigenstates are generally spin dependent, i.e., the spatial parts of the up and down natural orbitals form two different sets. A measure to quantify this spin dependence is introduced and it is shown that it varies by several orders of magnitude depending on the system. We also study the ordering issue of the spin-dependent occupation numbers which has practical implications in reduced density matrix functional theory minimization schemes, when generalized Pauli constraints (GPCs) are imposed and in the form of the CI expansion in terms of the natural orbitals. Finally, we discuss the aforementioned CI expansion when there are GPCs that are almost "pinned."
We investigate the structure of the one-body reduced density matrix of three electron systems, i.e., doublet and quadruplet spin configurations, corresponding to the smallest interacting system with an open-shell ground state. To this end, we use configuration interaction (CI) expansions of the exact wave function in Slater determinants built from natural orbitals in a finite dimensional Hilbert space. With the exception of maximally polarized systems, the natural orbitals of spin eigenstates are generally spin dependent, i.e., the spatial parts of the up and down natural orbitals form two different sets. A measure to quantify this spin dependence is introduced and it is shown that it varies by several orders of magnitude depending on the system. We also study the ordering issue of the spin-dependent occupation numbers which has practical implications in reduced density matrix functional theory minimization schemes, when generalized Pauli constraints (GPCs) are imposed and in the form of the CI expansion in terms of the natural orbitals. Finally, we discuss the aforementioned CI expansion when there are GPCs that are almost “pinned.”
A factorization of the wave-function coefficients obtained from configuration-interaction expansions is presented. For this purpose, the system orbitals are partitioned into a set of overlapping domains and a factor is associated with a given electron distribution in each domain. The configuration-interaction coefficient of each Slater determinant is given by the product of all the domain factors that are associated with this determinant. This expansion is asymptotically exact, which means that the exact wave function is recovered in the limit of large domains. Moreover, the number of factors needed to define a wave function has a slow growth as a function of the system size. The idea is illustrated through applications to two model systems, i.e. a collection of non-interacting electron pairs and a Néel-type wave function. A possible strategy for the calculation of the configuration factors is also briefly discussed.
We investigate whether the natural orbitals (NOs) minimize ∥Ψ−Φ∥2\lVert\Psi - \Phi\rVert^2∥Ψ−Φ∥2, where Ψ\PsiΨ is a wave function and Φ\PhiΦ is a full configuration interaction (CI) approximation to Ψ\PsiΨ in a truncated basis. We will show that the NOs rarely provide the optimal orbitals for Φ\PhiΦ, except when (1) there are only two particles or (2) only one basis function is removed in the case of fermions. Further, we will show that the CI expansion coefficients of Ψ\PsiΨ and Φ\PhiΦ are identical up to a global scaling factor and demonstrate how the NOs can be used to generate the orbitals that minimize ∥Ψ−Φ∥2\lVert\Psi - \Phi\rVert^2∥Ψ−Φ∥2.