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Physical Review A

Variational reduced-density-matrix theory applied to the electronic structure of few-electron quantum dots

Adam E. Rothman, David A. Mazziotti

Published Sep 9, 2008• 27 citations
Publisher

Abstract

Variational two-electron reduced-density-matrix (2RDM) theory is applied to computing energy spectra and properties of few-electron quantum dots. The model system is a two-dimensional electron gas with a central confinement potential. For each orbital angular momentum $J$, the energy and 2RDM are computed by the variational 2RDM method in which the energy is minimized as a functional of the 2RDM. In the minimization, which is performed by semidefinite programming, the 2RDM is constrained to represent a $N$-electron wave function with angular momentum $J$ by $N$- and $J$-representability conditions [D. A. Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)]. Advantages of the variational 2RDM method include (i) lower bounds on the energies for all $J$ values, (ii) calculation of approximate 2RDMs in polynomial time without many-electron wave functions, (iii) exploitation of angular symmetry in the sparse block-diagonal structure of the 2RDM, (iv) accurate description of multireference correlation (entanglement) effects, and (v) direct calculation of one- and two-electron properties from the 2RDM. With the 2RDM we directly compute pair-correlation functions, radial charge densities, and average radial electron displacements in the quantum dot. Energies and properties are compared to those from solving the $N$-electron Schr\"odinger equation by large-scale, exact diagonalization. It is found that the accuracy of the variational 2RDM approach is sensitive to the total orbital angular momentum and the symmetry of the final wave function. For quantum dots of high symmetry, the variational algorithm isolates a highly accurate solution that recovers the correlation energy within a few percent.

DOI
10.1103/physreva.78.032510

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Variational two-electron reduced-density-matrix (2RDM) theory is applied to computing energy spectra and properties of few-electron quantum dots. The model system is a two-dimensional electron gas with a central confinement potential. For each orbital angular momentum JJJ, the energy and 2RDM are computed by the variational 2RDM method in which the energy is minimized as a functional of the 2RDM. In the minimization, which is performed by semidefinite programming, the 2RDM is constrained to represent a NNN-electron wave function with angular momentum JJJ by NNN- and JJJ-representability conditions [D. A. Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)]. Advantages of the variational 2RDM method include (i) lower bounds on the energies for all JJJ values, (ii) calculation of approximate 2RDMs in polynomial time without many-electron wave functions, (iii) exploitation of angular symmetry in the sparse block-diagonal structure of the 2RDM, (iv) accurate description of multireference correlation (entanglement) effects, and (v) direct calculation of one- and two-electron properties from the 2RDM. With the 2RDM we directly compute pair-correlation functions, radial charge densities, and average radial electron displacements in the quantum dot. Energies and properties are compared to those from solving the NNN-electron Schr"odinger equation by large-scale, exact diagonalization. It is found that the accuracy of the variational 2RDM approach is sensitive to the total orbital angular momentum and the symmetry of the final wave function. For quantum dots of high symmetry, the variational algorithm isolates a highly accurate solution that recovers the correlation energy within a few percent.
The energy and properties of a many-electron atom or molecule may be directly computed from a variational optimization of a two-electron reduced density matrix (2RDM) that is constrained to represent many-electron quantum systems. In this paper we implement a variational 2RDM method with a representability constraint, known as the T2{T}_{2}T2​ condition. The optimization of the 2RDM is performed with a first-order algorithm for semidefinite programming [D. A. Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)] which, because of its lower computational cost in comparison to second-order methods, allows the treatment of larger basis sets. We also derive and implement a spin- and symmetry-adapted formulation of the T2{T}_{2}T2​ condition that significantly decreases the size of the largest block in the T2{T}_{2}T2​ matrix. The T2{T}_{2}T2​ condition, originally derived by Erdahl [Int. J. Quantum Chem. 13, 697 (1978)], was recently applied via a second-order algorithm to atoms and molecules [Z. Zhao et al., J. Chem. Phys. 120, 2095 (2004)]. While these calculations were restricted to molecules at equilibrium geometries in minimal basis sets, we apply the 2RDM method with the T2{T}_{2}T2​ condition to compute the electronic energies of molecules in both minimal and nonminimal basis sets at equilibrium as well as nonequilibrium geometries. Accurate potential energies curves are produced for BH, HF, and N2{\mathrm{N}}_{2}N2​. Results are compared with the 2RDM method without the T2{T}_{2}T2​ condition as well as several wave-function methods.
Variational reduced-density-matrix theory is applied to calculating the ground-state energy and two-electron reduced density matrices (2-RDMs) of the one-dimensional Hubbard model for a range of interaction strengths. The 2-RDM is constrained to represent an NNN-particle wave function by two sets of NNN-representability conditions, known as the 2- and partial 3-positivity conditions. Variational optimization of the energy with the 2-RDM constrained by NNN-representability conditions is performed using a first-order semidefinite-programming algorithm that was developed for treating atoms and molecules [D. A. Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)]. Accurate energies for a broad range of interaction strengths indicate that the variational 2-RDM method is a valuable tool for studying strongly correlated electrons.
Abstract The two‐electron reduced density matrix (2RDM) carries enough information to evaluate the electronic energy of a many‐electron system. The variational 2RDM (v2RDM) approach seeks to determine the 2RDM directly, without knowledge of the wave function, by minimizing this energy with respect to variations in the elements of the 2RDM, while also enforcing known N ‐representability conditions. In this tutorial review, we provide an overview of the theoretical underpinnings of the v2RDM approach and the N ‐representability constraints that are typically applied to the 2RDM. We also discuss the semidefinite programming (SDP) techniques used in v2RDM computations and provide enough Python code to develop a working v2RDM code that interfaces to the libSDP library of SDP solvers. This article is categorized under: Electronic Structure Theory > Ab Initio Electronic Structure Methods Software > Quantum Chemistry
The last several decades have seen significant advances in the theoretical modeling of materials within the fields of solid-state physics and materials science, but many methods commonly applied to this problem struggle to capture strong electron correlation accurately. Recent widespread interest in quantum materials---where strong correlation plays a crucial role in the quantum effects governing their behavior---further highlights the need for theoretical methods capable of rigorously treating such correlation. Here, we present a periodic generalization of variational two-electron reduced density matrix (2-RDM) theory, a bootstrapping-type method that minimizes the ground-state energy as a functional of the 2-RDM without relying on the wave function. The 2-RDM is computed directly by semidefinite programming with NNN-representability conditions, ensuring accurate treatment of strongly correlated electronic systems. By exploiting translational symmetry, we significantly reduce computational scaling, enabling applications to realistic materials-scale systems. Additionally, we introduce an alternative to conventional energy band structures: natural-orbital occupation-number bands, which, being independent of mean-field assumptions, offer deeper insights into electron correlation effects. We demonstrate the effectiveness of this approach by applying the theory to hydrogen chains, molybdenum disulfide, and nickel oxide, showing that natural-orbital occupation bands correctly capture electronic character in regimes where density functional theory fails. This work represents a major step toward accurately describing the electronic structure of quantum materials using reduced density matrices rather than wave functions.
For 50 years, progress toward the direct calculation of the ground-state two-electron reduced density matrix (2-RDM) was stymied from an inability to constrain the 2-RDM to represent an N-electron wave function. Recent advances in theory and optimization have realized the direct calculation of the 2-RDM. A variational 2-RDM procedure, using first-order semidefinite programming, has been shown to capture multireference correlation effects important at nonequilibrium geometries [Mazziotti, Phys. Rev. Lett. 2004, 93, No. 213001]. This method emerged from research on a nonvariational calculation of the 2-RDM by the contracted Schrödinger equation. Both approaches will be discussed and illustrated.
The variational two-electron reduced-density-matrix (2-RDM) method computes the ground-state energy and 2-RDM of an atom or molecule without calculation of the many-electron wave function. Recently, the computational efficiency of the 2-RDM method has been significantly enhanced through the use of a first-order algorithm for semidefinite programming [Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)]. In this paper we develop a spin- and symmetry-adapted formulation of the method that further improves its efficiency by incorporating both the spin and spatial symmetries of many-electron atoms and molecules. While previous work on density-matrix symmetry focused on only one form of the 2-RDM, the variational method employs three different forms of the 2-RDM, known as the DDD, QQQ, and GGG matrices, to restrict the 2-RDM to be approximately NNN-representable, that is representable by an NNN-electron wave function. We apply spin symmetries to the three forms of the 2-RDM, each of which breaks into four diagonal spin-blocks, namely one singlet and three triplet blocks. If the molecules have point-group symmetry, each of the 2-RDMs may be further subdivided into smaller diagonal blocks according to the spatial symmetry of the basis functions. The subdivision of the 2-RDMs into diagonal blocks generates significant computational savings in both floating-point operations and memory storage. Calculations illustrate the computational savings. Spin adaptation also enforces the correct expectation value of the ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{S}}^{2}$ operator, which in earlier work is applied as a separate constraint.