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
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 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 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 condition that significantly decreases the size of the largest block in the T2 matrix. The 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 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. Results are compared with the 2RDM method without the T2 condition as well as several wave-function methods.
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.
The reduced density matrix (RDM) method, which is a variational calculation based on the second-order reduced density matrix, is applied to the ground state energies and the dipole moments for 57 different states of atoms, molecules, and to the ground state energies and the elements of 2-RDM for the Hubbard model. We explore the well-known N-representability conditions (P, Q, and G) together with the more recent and much stronger T1 and T2(') conditions. T2(') condition was recently rederived and it implies T2 condition. Using these N-representability conditions, we can usually calculate correlation energies in percentage ranging from 100% to 101%, whose accuracy is similar to CCSD(T) and even better for high spin states or anion systems where CCSD(T) fails. Highly accurate calculations are carried out by handling equality constraints and/or developing multiple precision arithmetic in the semidefinite programming (SDP) solver. Results show that handling equality constraints correctly improves the accuracy from 0.1 to 0.6 mhartree. Additionally, improvements by replacing T2 condition with T2(') condition are typically of 0.1-0.5 mhartree. The newly developed multiple precision arithmetic version of SDP solver calculates extraordinary accurate energies for the one dimensional Hubbard model and Be atom. It gives at least 16 significant digits for energies, where double precision calculations gives only two to eight digits. It also provides physically meaningful results for the Hubbard model in the high correlation limit.
The variational two-electron reduced density matrix (v2RDM) method is generalized for the description of total angular momentum ( J ) and projection of total angular momentum ( M J ) states in atomic systems described by nonrelativistic Hamiltonians, and it is shown that the approach exhibits serious deficiencies. Under ensemble N -representability constraints, v2RDM theory fails to retain the appropriate degeneracies among various J states for fixed spin ( S ) and orbital angular momentum ( L ), and for fixed L, S, and J, the manifold of M J states is not necessarily degenerate. Moreover, a substantial energy error is observed for a system for which the two-electron reduced density matrix is exactly ensemble N -representable; in this case, the error stems from violations in pure-state N -representability conditions. Unfortunately, such violations do not appear to be good indicators of the reliability of energies from v2RDM theory in general. Several states are identified for which energy errors are near zero and yet pure-state conditions are clearly violated.
Abstract : Variational two-electron reduced-density-matrix (2-RDM) methods provide a reference independent description of the electronic structure ofmany-electron systems; these methods also naturally capture multireference correlation effects. However, existing implementations of thevariational 2-RDM (v-2RDM) method exhibit a time-to-solution that is often prohibitively large for the general description of largemolecules or materials. It is the goal of this STTR to develop an efficient, parallel, and commercializable 2-RDM-based electronic structure software suite. In Phase I of this STTR, we have: (i) developed a prototype commercializable software product to perform electronicstructure computations using the variational 2-RDM method, (ii) developed two semidefinite programming solvers and identified which ismore likely to be of use for solving the variational 2-RDM problem for general systems, (iii) interfaced this solver with the libraries of theQ-Chem electronic structure package, (iv) implemented algorithms that use either loop-based or libtensor-based tensor manipulation andidentified the libtensor algorithm as the more promising for massively parallel variational 2-RDM computations, (v) developed a pilotdistributed-memory parallel version of the solver through an interface between libtensor and the Cyclops Tensor Framework (CTF), (vi)enabled the automatic detection and use of graphics processing units (GPUs) for certain portions of the algorithm, and (vii) extended someof our algorithms for the description of open-shell systems and to include partial three-particle N-representability conditions.
Direct variational calculation of two-electron reduced density matrices (2-RDMs) for many-electron atoms and molecules in nonminimal basis sets has recently been achieved through the use of first-order semidefinite programming [D. A. Mazziotti, Phys. Rev. Lett. (in press)]. With semidefinite programming, the electronic ground-state energy of a molecule is minimized with respect to the 2-RDM subject to N-representability constraints known as positivity conditions. Here we present a detailed account of the first-order algorithm for semidefinite programming and its comparison with the primal-dual interior-point algorithms employed in earlier variational 2-RDM calculations. The first-order semidefinite-programming algorithm, computations show, offers an orders-of-magnitude reduction in floating-point operations and storage in comparison with previous implementations. We also examine the ability of the positivity conditions to treat strong correlation and multireference effects through an analysis of the Hamiltonians for which the conditions are exact. Calculations are performed in nonminimal basis sets for a variety of atoms and molecules and the potential-energy curves for CO and H(2)O.