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

The Treatment of Diradical Systems and Molecular Optimizations with Two-Electron Reduced Density Matrices

Andrew J. Valentine

Published Jan 1, 2017• 0 citations
Publisher

Abstract

The electronic Hamiltonian contains only pairwise interactions, allowing the energy of an electronic system to be expressed in terms of the two-electron reduced-density-matrix (2-RDM) in lieu of the many-electron wavefunction. The variable space for the exact N-electron wavefunction scales exponentially with the size of the system, while the 2-RDM is polynomial in scale. By using the 2-RDM as the primary variable in electronic structure calculations, it may be possible to obtain very accurate energies at a much more favorable scaling than wavefunction methods. In this thesis, we will use two existing 2-RDM methods to treat electronic systems. First, we will apply the active-space variational 2-RDM method, which directly minimizes the energy with respect to the 2-RDM, to a cadmium telluride polymer that was recently used to greatly enhance the conductivity of CdTe quantum dots. We find that this polymer is very highly correlated despite a deceptively simple structure. We will then turn to the parametric 2-RDM method (p2-RDM), which parameterizes the 2-RDM in terms of a truncated configuration interaction ansatz, but which includes additional flexibility in order to be size-extensive. We apply p2-RDM to the study of the olympicene molecule, which features both fully aromatic and diradical isomers. The parametric 2-RDM method predicts that all isomers are stable to dissociation, in contrast to coupled cluster methods which do not predict stable diradical states. We then present analytical nuclear gradients for p2-RDM, which greatly decrease the number of calculations required to perform geometry optimizations. We apply these gradients to the study of trans-polyacetylene, for which p2-RDM, unlike many wavefunction methods, is able to predict a bond length alternation (BLA) to within experimental values. Lastly, as a single-reference method, p2-RDM may encounter numerical difficulties when the reference wavefunction is of particularly poor quality. We propose a modification to the parameterization that may render the method more generally robust.

DOI
10.6082/m18w3bdx

Related papers

Pair 2-electron reduced density matrix theory using localized orbitals

Kade Head-Marsden, David A. Mazziotti

Full configuration interaction (FCI) restricted to a pairing space yields size-extensive correlation energies but its cost scales exponentially with molecular size. Restricting the variational two-electron reduced-density-matrix (2-RDM) method to represent the same pairing space yields an accurate lower bound to the pair FCI energy at a mean-field-like computational scaling of O(r3) where r is the number of orbitals. In this paper, we show that localized molecular orbitals can be employed to generate an efficient, approximately size-extensive pair 2-RDM method. The use of localized orbitals eliminates the substantial cost of optimizing iteratively the orbitals defining the pairing space without compromising accuracy. In contrast to the localized orbitals, the use of canonical Hartree-Fock molecular orbitals is shown to be both inaccurate and non-size-extensive. The pair 2-RDM has the flexibility to describe the spectra of one-electron RDM occupation numbers from all quantum states that are invariant to time-reversal symmetry. Applications are made to hydrogen chains and their dissociation, n-acene from naphthalene through octacene, and cadmium telluride 2-, 3-, and 4-unit polymers. For the hydrogen chains, the pair 2-RDM method recovers the majority of the energy obtained from similar calculations that iteratively optimize the orbitals. The localized-orbital pair 2-RDM method with its mean-field-like computational scaling and its ability to describe multi-reference correlation has important applications to a range of strongly correlated phenomena in chemistry and physics.

The Journal of Chemical Physics•Aug 22, 2017•37 citations

Parametrization of the two-electron reduced density matrix for its direct calculation without the many-electron wave function: Generalizations and applications

David A. Mazziotti

An improved parametrization of the two-electron reduced density matrix (2-RDM) [D. A. Mazziotti, Phys. Rev. Lett. 101, 253002 (2008)] was recently shown to yield energies and properties that are markedly better than those calculated by traditional ab initio methods of similar computational scaling. In this paper a family of such energy functionals, generalizing the ones obtained previously, is derived through the use of (i) $p$-particle contraction relations based on the contraction of the cumulant expansions of $p$-particle RDMs and (ii) Cauchy-Schwarz relations that arise from an important set of $N$-representability constraints known as the two-positivity conditions. The 2-RDMs are explicitly parameterized in terms of the first-order part of the cumulant 2-RDM and, for the inclusion of single excitations, a second-order part of the 1-RDM. In contrast to earlier formulations based on the coefficients from configuration interaction with single and double excitations (CISD), the cumulant-based parametric 2-RDM methods, from the properties of cumulants, are rigorously size extensive. We also show that writing the energy functionals in terms of correlated 1-RDMs and cumulant 2-RDMs reduces the computational cost of the parametric 2-RDM methods to that of CISD. Applications are made to ground-state energies of several molecules, equilibrium bond distances, and frequencies of HF, ${\mathrm{F}}_{2}$, and CO, the relative energy of the cis and trans isomers of ${\mathrm{HO}}_{3}$${}^{\ensuremath{-}}$, and the HCN-HNC isomerization reaction. For bond breaking in hydrogen fluoride the improved and more efficient parametric 2-RDM methods yield energies with similar accuracies at both equilibrium and nonequilibrium geometries in 6-31G** and polarized valence quadruple-$\ensuremath{\zeta}$ basis sets. Computed 2-RDMs very nearly satisfy well-known $N$-representability conditions.

Physical Review A•Jun 30, 2010•52 citations

Reduced Density Matrices and Tensor Factorizations for Low Cost Quantum Chemistry

Erik P. Hoy

Conventional low cost computational methods in quantum chemistry such as Hartree-Fock and density functional theory often fail to properly describe electron correlation. Of the two types of electron correlation, dynamic and static, static or multi-reference correlation is particularly difficult to obtain using low cost methods. This type of electron correlation results when multiple Hartree-Fock reference states are needed to describe a system. One class of methods which are able to describe both types effectively are 2-electron reduced density matrix (2-RDM) methods. These methods use the 2-electron density matrix as the primary variable in the electronic Schrodinger equation in place of the wavefunction. By circumventing the wavefunction, one replaces an exponentially scaling problem with a polynomially scaling one, and unlike density functional theory, the energy is a linear functional of the 2-RDM. The lowest cost 2-RDM method currently available is known as the parametric 2-electron reduced density matrix method (p2-RDM). The method has equivalent scaling to configuration interaction with single and double excitations (CISD) and is able to treat both dynamic and static correlation. This work examines both the correlation recovery abilities of the p2-RDM method and potential scaling improvements to the method using tensor factorization. We first investigated several molecular systems including the conversion of hydroxylamine to ammonia oxide, the hydridotrioxygen atmospheric radical, and the benzene and cyclobutadiene diradicals. The purpose of these studies was to further clarify the ability of the parametric method to capture various mixes of both types of correlation. We then investigated using two different tensor factorizations, low rank spectral expansion and tensor hypercontraction, to reduce the scaling of the p2-RDM toward that of Hartree-Fock and density functional theory. Based on these results, we developed a new tensor factorization to address the shortcomings of previous methods and provide a promising, new technique to generate a fourth order p2-RDM method capable of capturing both dynamic and static correlation.

•Jan 1, 2015•0 citations

Active-space two-electron reduced-density-matrix method: Complete active-space calculations without diagonalization of the N-electron Hamiltonian

Gergely Gidofalvi, David A. Mazziotti

Molecular systems in chemistry often have wave functions with substantial contributions from two-or-more electronic configurations. Because traditional complete-active-space self-consistent-field (CASSCF) methods scale exponentially with the number N of active electrons, their applicability is limited to small active spaces. In this paper we develop an active-space variational two-electron reduced-density-matrix (2-RDM) method in which the expensive diagonalization is replaced by a variational 2-RDM calculation where the 2-RDM is constrained by approximate N-representability conditions. Optimization of the constrained 2-RDM is accomplished by large-scale semidefinite programming [Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)]. Because the computational cost of the active-space 2-RDM method scales polynomially as r(a)(6) where r(a) is the number of active orbitals, the method can be applied to treat active spaces that are too large for conventional CASSCF. The active-space 2-RDM method performs two steps: (i) variational calculation of the 2-RDM in the active space and (ii) optimization of the active orbitals by Jacobi rotations. For large basis sets this two-step 2-RDM method is more efficient than the one-step, low-rank variational 2-RDM method [Gidofalvi and Mazziotti, J. Chem. Phys. 127, 244105 (2007)]. Applications are made to HF, H(2)O, and N(2) as well as n-acene chains for n=2-8. When n>4, the acenes cannot be treated by conventional CASSCF methods; for example, when n=8, CASSCF requires optimization over approximately 1.47x10(17) configuration state functions. The natural occupation numbers of the n-acenes show the emergence of bi- and polyradical character with increasing chain length.

The Journal of Chemical Physics•Oct 3, 2008•187 citations

THE VARIATIONAL TWO-ELECTRON REDUCED-DENSITY-MATRIX METHOD FOR EXTENDED SYSTEMS

Nicholas C. Rubin

In this thesis we develop the variational two-electron reduced-density-matrix method for extended systems. Extended systems are represented in two ways: i) lattice models describing the dominant valence electronic structure with periodic boundaries to account for their extended nature and ii) a crystalline-orbital basis built from atomic orbitals using the generalization of molecular orbital theory to polymers. The first part of this thesis (Ch. 3-4) examines the performance of the variational 2-RDM method on lattice systems with tunable electron correlation. The first of these systems is the classic Hubbard model with linear and ladder lattice topologies. Because electron correlation functions, such as charge- and spin-ordering, are linear functions of the 2-RDM, the difference in electronic structure between one- and quasi-one-dimensional systems is accurately characterized. The second model contains only two-body interactions and is unique among typical spin models in that it does not have a mean-field reference wave function. The ground state wave functions from all Hamiltonians in the model have the same 1-electron reduced density matrix; consequently, one-electron theories are largely inapplicable. The superconducting eta-pairing ground states make the model a unique tool for demonstrating the necessary N-representability in highly correlated environments. The second part of this thesis (Ch. 5-6) develops a formalism for modeling materials by solving the full Schrodinger equation. Crystalline-orbital Hartree-Fock provides a set of orbitals and integral tensors for the variational 2-RDM method. We demonstrate that time-reversal symmetry, which is implicitly included in position space electronic structure calculations, must be explicitly included as an N-representability constraint on the 2-RDM when using a momentum space basis. The necessity of these equality constraints is demonstrated by the accurate recovery of the binding energy of two polymers and the symmetry of their natural orbital occupations.

•Jan 1, 2016•0 citations

First-order semidefinite programming for the two-electron treatment of many-electron atoms and molecules

David A. Mazziotti

The ground-state energy and properties of any many-electron atom or molecule may be rigorously computed by variationally computing the two-electron reduced density matrix rather than the many-electron wavefunction. While early attempts fifty years ago to compute the ground-state 2-RDM directly were stymied because the 2-RDM must be constrained to represent an N-electron wavefunction, recent advances in theory and optimization have made direct computation of the 2-RDM possible. The constraints in the variational calculation of the 2-RDM require a special optimization known as a semidefinite programming. Development of first-order semidefinite programming for the 2-RDM method has reduced the computational costs of the calculation by orders of magnitude [Mazziotti, Phys. Rev. Lett. 93 (2004) 213001]. The variational 2-RDM approach is effective at capturing multi-reference correlation effects that are especially important at non-equilibrium molecular geometries. Recent work on 2-RDM methods will be reviewed and illustrated with particular emphasis on the importance of advances in large-scale semidefinite programming.

ESAIM Mathematical Modelling and Numerical Analysis•Mar 1, 2007•35 citations
The electronic Hamiltonian contains only pairwise interactions, allowing the energy of an electronic system to be expressed in terms of the two-electron reduced-density-matrix (2-RDM) in lieu of the many-electron wavefunction. The variable space for the exact N-electron wavefunction scales exponentially with the size of the system, while the 2-RDM is polynomial in scale. By using the 2-RDM as the primary variable in electronic structure calculations, it may be possible to obtain very accurate energies at a much more favorable scaling than wavefunction methods. In this thesis, we will use two existing 2-RDM methods to treat electronic systems. First, we will apply the active-space variational 2-RDM method, which directly minimizes the energy with respect to the 2-RDM, to a cadmium telluride polymer that was recently used to greatly enhance the conductivity of CdTe quantum dots. We find that this polymer is very highly correlated despite a deceptively simple structure. We will then turn to the parametric 2-RDM method (p2-RDM), which parameterizes the 2-RDM in terms of a truncated configuration interaction ansatz, but which includes additional flexibility in order to be size-extensive. We apply p2-RDM to the study of the olympicene molecule, which features both fully aromatic and diradical isomers. The parametric 2-RDM method predicts that all isomers are stable to dissociation, in contrast to coupled cluster methods which do not predict stable diradical states. We then present analytical nuclear gradients for p2-RDM, which greatly decrease the number of calculations required to perform geometry optimizations. We apply these gradients to the study of trans-polyacetylene, for which p2-RDM, unlike many wavefunction methods, is able to predict a bond length alternation (BLA) to within experimental values. Lastly, as a single-reference method, p2-RDM may encounter numerical difficulties when the reference wavefunction is of particularly poor quality. We propose a modification to the parameterization that may render the method more generally robust.
Full configuration interaction (FCI) restricted to a pairing space yields size-extensive correlation energies but its cost scales exponentially with molecular size. Restricting the variational two-electron reduced-density-matrix (2-RDM) method to represent the same pairing space yields an accurate lower bound to the pair FCI energy at a mean-field-like computational scaling of O(r3) where r is the number of orbitals. In this paper, we show that localized molecular orbitals can be employed to generate an efficient, approximately size-extensive pair 2-RDM method. The use of localized orbitals eliminates the substantial cost of optimizing iteratively the orbitals defining the pairing space without compromising accuracy. In contrast to the localized orbitals, the use of canonical Hartree-Fock molecular orbitals is shown to be both inaccurate and non-size-extensive. The pair 2-RDM has the flexibility to describe the spectra of one-electron RDM occupation numbers from all quantum states that are invariant to time-reversal symmetry. Applications are made to hydrogen chains and their dissociation, n-acene from naphthalene through octacene, and cadmium telluride 2-, 3-, and 4-unit polymers. For the hydrogen chains, the pair 2-RDM method recovers the majority of the energy obtained from similar calculations that iteratively optimize the orbitals. The localized-orbital pair 2-RDM method with its mean-field-like computational scaling and its ability to describe multi-reference correlation has important applications to a range of strongly correlated phenomena in chemistry and physics.
An improved parametrization of the two-electron reduced density matrix (2-RDM) [D. A. Mazziotti, Phys. Rev. Lett. 101, 253002 (2008)] was recently shown to yield energies and properties that are markedly better than those calculated by traditional ab initio methods of similar computational scaling. In this paper a family of such energy functionals, generalizing the ones obtained previously, is derived through the use of (i) ppp-particle contraction relations based on the contraction of the cumulant expansions of ppp-particle RDMs and (ii) Cauchy-Schwarz relations that arise from an important set of NNN-representability constraints known as the two-positivity conditions. The 2-RDMs are explicitly parameterized in terms of the first-order part of the cumulant 2-RDM and, for the inclusion of single excitations, a second-order part of the 1-RDM. In contrast to earlier formulations based on the coefficients from configuration interaction with single and double excitations (CISD), the cumulant-based parametric 2-RDM methods, from the properties of cumulants, are rigorously size extensive. We also show that writing the energy functionals in terms of correlated 1-RDMs and cumulant 2-RDMs reduces the computational cost of the parametric 2-RDM methods to that of CISD. Applications are made to ground-state energies of several molecules, equilibrium bond distances, and frequencies of HF, F2{\mathrm{F}}_{2}F2​, and CO, the relative energy of the cis and trans isomers of HO3{\mathrm{HO}}_{3}HO3​${}^{\ensuremath{-}}$, and the HCN-HNC isomerization reaction. For bond breaking in hydrogen fluoride the improved and more efficient parametric 2-RDM methods yield energies with similar accuracies at both equilibrium and nonequilibrium geometries in 6-31G** and polarized valence quadruple-$\ensuremath{\zeta}$ basis sets. Computed 2-RDMs very nearly satisfy well-known NNN-representability conditions.
Conventional low cost computational methods in quantum chemistry such as Hartree-Fock and density functional theory often fail to properly describe electron correlation. Of the two types of electron correlation, dynamic and static, static or multi-reference correlation is particularly difficult to obtain using low cost methods. This type of electron correlation results when multiple Hartree-Fock reference states are needed to describe a system. One class of methods which are able to describe both types effectively are 2-electron reduced density matrix (2-RDM) methods. These methods use the 2-electron density matrix as the primary variable in the electronic Schrodinger equation in place of the wavefunction. By circumventing the wavefunction, one replaces an exponentially scaling problem with a polynomially scaling one, and unlike density functional theory, the energy is a linear functional of the 2-RDM. The lowest cost 2-RDM method currently available is known as the parametric 2-electron reduced density matrix method (p2-RDM). The method has equivalent scaling to configuration interaction with single and double excitations (CISD) and is able to treat both dynamic and static correlation. This work examines both the correlation recovery abilities of the p2-RDM method and potential scaling improvements to the method using tensor factorization. We first investigated several molecular systems including the conversion of hydroxylamine to ammonia oxide, the hydridotrioxygen atmospheric radical, and the benzene and cyclobutadiene diradicals. The purpose of these studies was to further clarify the ability of the parametric method to capture various mixes of both types of correlation. We then investigated using two different tensor factorizations, low rank spectral expansion and tensor hypercontraction, to reduce the scaling of the p2-RDM toward that of Hartree-Fock and density functional theory. Based on these results, we developed a new tensor factorization to address the shortcomings of previous methods and provide a promising, new technique to generate a fourth order p2-RDM method capable of capturing both dynamic and static correlation.
Molecular systems in chemistry often have wave functions with substantial contributions from two-or-more electronic configurations. Because traditional complete-active-space self-consistent-field (CASSCF) methods scale exponentially with the number N of active electrons, their applicability is limited to small active spaces. In this paper we develop an active-space variational two-electron reduced-density-matrix (2-RDM) method in which the expensive diagonalization is replaced by a variational 2-RDM calculation where the 2-RDM is constrained by approximate N-representability conditions. Optimization of the constrained 2-RDM is accomplished by large-scale semidefinite programming [Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)]. Because the computational cost of the active-space 2-RDM method scales polynomially as r(a)(6) where r(a) is the number of active orbitals, the method can be applied to treat active spaces that are too large for conventional CASSCF. The active-space 2-RDM method performs two steps: (i) variational calculation of the 2-RDM in the active space and (ii) optimization of the active orbitals by Jacobi rotations. For large basis sets this two-step 2-RDM method is more efficient than the one-step, low-rank variational 2-RDM method [Gidofalvi and Mazziotti, J. Chem. Phys. 127, 244105 (2007)]. Applications are made to HF, H(2)O, and N(2) as well as n-acene chains for n=2-8. When n>4, the acenes cannot be treated by conventional CASSCF methods; for example, when n=8, CASSCF requires optimization over approximately 1.47x10(17) configuration state functions. The natural occupation numbers of the n-acenes show the emergence of bi- and polyradical character with increasing chain length.
In this thesis we develop the variational two-electron reduced-density-matrix method for extended systems. Extended systems are represented in two ways: i) lattice models describing the dominant valence electronic structure with periodic boundaries to account for their extended nature and ii) a crystalline-orbital basis built from atomic orbitals using the generalization of molecular orbital theory to polymers. The first part of this thesis (Ch. 3-4) examines the performance of the variational 2-RDM method on lattice systems with tunable electron correlation. The first of these systems is the classic Hubbard model with linear and ladder lattice topologies. Because electron correlation functions, such as charge- and spin-ordering, are linear functions of the 2-RDM, the difference in electronic structure between one- and quasi-one-dimensional systems is accurately characterized. The second model contains only two-body interactions and is unique among typical spin models in that it does not have a mean-field reference wave function. The ground state wave functions from all Hamiltonians in the model have the same 1-electron reduced density matrix; consequently, one-electron theories are largely inapplicable. The superconducting eta-pairing ground states make the model a unique tool for demonstrating the necessary N-representability in highly correlated environments. The second part of this thesis (Ch. 5-6) develops a formalism for modeling materials by solving the full Schrodinger equation. Crystalline-orbital Hartree-Fock provides a set of orbitals and integral tensors for the variational 2-RDM method. We demonstrate that time-reversal symmetry, which is implicitly included in position space electronic structure calculations, must be explicitly included as an N-representability constraint on the 2-RDM when using a momentum space basis. The necessity of these equality constraints is demonstrated by the accurate recovery of the binding energy of two polymers and the symmetry of their natural orbital occupations.
The ground-state energy and properties of any many-electron atom or molecule may be rigorously computed by variationally computing the two-electron reduced density matrix rather than the many-electron wavefunction. While early attempts fifty years ago to compute the ground-state 2-RDM directly were stymied because the 2-RDM must be constrained to represent an N-electron wavefunction, recent advances in theory and optimization have made direct computation of the 2-RDM possible. The constraints in the variational calculation of the 2-RDM require a special optimization known as a semidefinite programming. Development of first-order semidefinite programming for the 2-RDM method has reduced the computational costs of the calculation by orders of magnitude [Mazziotti, Phys. Rev. Lett. 93 (2004) 213001]. The variational 2-RDM approach is effective at capturing multi-reference correlation effects that are especially important at non-equilibrium molecular geometries. Recent work on 2-RDM methods will be reviewed and illustrated with particular emphasis on the importance of advances in large-scale semidefinite programming.