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Verhandlungen der Deutschen Physikalischen Gesellschaft

Effective Hamiltonian for the 1-body reduced density matrix functional theory

Nektarios N. Lathiotakis, N. Helbig, Eberhard K. U. Gross

Published Jan 1, 2005• 0 citations
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Nonorthogonal orbital based N-body reduced density matrices and their applications to valence bond theory. I. Hamiltonian matrix elements between internally contracted excited valence bond wave functions

Zhenhua Chen, Xun Chen, Wei Wu

In this series, the n-body reduced density matrix (n-RDM) approach for nonorthogonal orbitals and their applications to ab initio valence bond (VB) methods are presented. As the first paper of this series, Hamiltonian matrix elements between internally contracted VB wave functions are explicitly provided by means of nonorthogonal orbital based RDM approach. To this end, a more generalized Wick's theorem, called enhanced Wick's theorem, is presented both in arithmetical and in graphical forms, by which the deduction of expressions for the matrix elements between internally contracted VB wave functions is dramatically simplified, and the matrix elements are finally expressed in terms of tensor contractions of electronic integrals and n-RDMs of the reference VB self-consistent field wave function. A string-based algorithm is developed for the purpose of evaluating n-RDMs in an efficient way. Using the techniques presented in this paper, one is able to develop new methods and efficient algorithms for nonorthogonal orbital based many-electron theory much easier than by use of the first quantized formulism.

The Journal of Chemical Physics•Apr 26, 2013•39 citations

One-body reduced density matrix functionals for the homogeneous electron gas

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Determining the N-Representability of a Reduced Density Matrix via Unitary Evolution and Stochastic Sampling

Gustavo E. Massaccesi, Ofelia B. Oña, P. Capuzzi +6 more

High Resolution Image Download MS PowerPoint Slide The N -representability problem consists in determining whether, for a given p -body matrix, there exists at least one N -body density matrix from which the p -body matrix can be obtained by contraction, that is, if the given matrix is a p -body reduced density matrix ( p -RDM). The knowledge of all necessary and sufficient conditions for a p -body matrix to be N -representable allows the constrained minimization of a many-body Hamiltonian expectation value with respect to the p -body density matrix and, thus, the determination of its exact ground state. However, the number of constraints that complete the N -representability conditions grows exponentially with system size, and hence, the procedure quickly becomes intractable for practical applications. This work introduces a hybrid quantum-stochastic algorithm to effectively replace the N -representability conditions. The algorithm consists of applying to an initial N -body density matrix a sequence of unitary evolution operators constructed from a stochastic process that successively approaches the reduced state of the density matrix on a p -body subsystem, represented by a p -RDM, to a target p -body matrix, potentially a p -RDM. The generators of the evolution operators follow the well-known adaptive derivative-assembled pseudo-Trotter method (ADAPT), while the stochastic component is implemented by using a simulated annealing process. The resulting algorithm is independent of any underlying Hamiltonian, and it can be used to decide whether a given p -body matrix is N -representable, establishing a criterion to determine its quality and correcting it. We apply the proposed hybrid ADAPT algorithm to alleged reduced density matrices from a quantum chemistry electronic Hamiltonian, from the reduced Bardeen–Cooper–Schrieffer model with constant pairing, and from the Heisenberg XXZ spin model. In all cases, the proposed method behaves as expected for 1-RDMs and 2-RDMs, evolving the initial matrices toward different targets.

arXiv:2503.17303•Journal of Chemical Theory and Computation•Nov 14, 2024•5 citations
In this series, the n-body reduced density matrix (n-RDM) approach for nonorthogonal orbitals and their applications to ab initio valence bond (VB) methods are presented. As the first paper of this series, Hamiltonian matrix elements between internally contracted VB wave functions are explicitly provided by means of nonorthogonal orbital based RDM approach. To this end, a more generalized Wick's theorem, called enhanced Wick's theorem, is presented both in arithmetical and in graphical forms, by which the deduction of expressions for the matrix elements between internally contracted VB wave functions is dramatically simplified, and the matrix elements are finally expressed in terms of tensor contractions of electronic integrals and n-RDMs of the reference VB self-consistent field wave function. A string-based algorithm is developed for the purpose of evaluating n-RDMs in an efficient way. Using the techniques presented in this paper, one is able to develop new methods and efficient algorithms for nonorthogonal orbital based many-electron theory much easier than by use of the first quantized formulism.
High Resolution Image Download MS PowerPoint Slide The N -representability problem consists in determining whether, for a given p -body matrix, there exists at least one N -body density matrix from which the p -body matrix can be obtained by contraction, that is, if the given matrix is a p -body reduced density matrix ( p -RDM). The knowledge of all necessary and sufficient conditions for a p -body matrix to be N -representable allows the constrained minimization of a many-body Hamiltonian expectation value with respect to the p -body density matrix and, thus, the determination of its exact ground state. However, the number of constraints that complete the N -representability conditions grows exponentially with system size, and hence, the procedure quickly becomes intractable for practical applications. This work introduces a hybrid quantum-stochastic algorithm to effectively replace the N -representability conditions. The algorithm consists of applying to an initial N -body density matrix a sequence of unitary evolution operators constructed from a stochastic process that successively approaches the reduced state of the density matrix on a p -body subsystem, represented by a p -RDM, to a target p -body matrix, potentially a p -RDM. The generators of the evolution operators follow the well-known adaptive derivative-assembled pseudo-Trotter method (ADAPT), while the stochastic component is implemented by using a simulated annealing process. The resulting algorithm is independent of any underlying Hamiltonian, and it can be used to decide whether a given p -body matrix is N -representable, establishing a criterion to determine its quality and correcting it. We apply the proposed hybrid ADAPT algorithm to alleged reduced density matrices from a quantum chemistry electronic Hamiltonian, from the reduced Bardeen–Cooper–Schrieffer model with constant pairing, and from the Heisenberg XXZ spin model. In all cases, the proposed method behaves as expected for 1-RDMs and 2-RDMs, evolving the initial matrices toward different targets.