Spatial and spin symmetry breaking in semidefinite-programming-based Hartree-Fock theory

The Hartree-Fock problem was recently recast as a semidefinite optimization over the space of rank-constrained two-body reduced-density matrices (RDMs) [Phys. Rev. A 89, 010502(R) (2014)]. This formulation of the problem transfers the non-convexity of the Hartree-Fock energy functional to the rank constraint on the two-body RDM. We consider an equivalent optimization over the space of positive semidefinite one-electron RDMs (1-RDMs) that retains the non-convexity of the Hartree-Fock energy expression. The optimized 1-RDM satisfies ensemble $N$-representability conditions, and ensemble spin-state conditions may be imposed as well. The spin-state conditions place additional linear and nonlinear constraints on the 1-RDM. We apply this RDM-based approach to several molecular systems and explore its spatial (point group) and spin ($S^2$ and $S_3$) symmetry breaking properties. When imposing $S^2$ and $S_3$ symmetry but relaxing point group symmetry, the procedure often locates spatial-symmetry-broken solutions that are difficult to identify using standard algorithms. For example, the RDM-based approach yields a smooth, spatial-symmetry-broken potential energy curve for the well-known Be--H$_2$ insertion pathway. We also demonstrate numerically that, upon relaxation of $S^2$ and $S_3$ symmetry constraints, the RDM-based approach is equivalent to real-valued generalized Hartree-Fock theory.Comment: 9 pages, 6 figure

Long-Tail Distributions And Total Returns On Risk-Based Investment Products

Use of target date funds (TDFs) in retirement plans increased in popularity following the 2007-2008 financial crisis. However, some argued that TDFs do not provide an acceptable level of protection against market downturns and long-tail events. This study assesses the ability of TDFs to deal with long-tail events. It builds a system of equations for seven asset classes that are used to build four hypothetical TDFs, and compares simulated total returns for four hypothetical TDFs over a 50-year horizon, where the stochastic terms for each of the underlying asset classes is based on the normal distribution and a long-tail distributions (the Laplace distribution). Simulations are repeated 2000 times. Total returns for four TDFs are calculated over non-overlapping 1-, 2- 3-, and 5-year horizons over the 50-year span for the 2000 simulations. Results show that about half the time the risk measure for the long-tail distribution are wider than the normal, and about half the time the opposite holds. It seems that the processes of asset diversification along with calculating returns over horizons of at least one year mitigates effects of long-tail characteristics. Even so, results do not bode well for those nearing retirement. Negative returns over 1- and 2-year investment horizons are possible 17% of the time for a conservative allocation over a 50-year period

IL1. Quantum Chemistry without Wavefunctions

Complete active space self-consistent field (CASSCF) methods are enormously important in quantum chemistry, as they allow for the description of states dominated by more than one electronic configuration. In CASSCF, the electronic structure of the active space is often represented by a configuration interaction (CI) expansion of the active-space wave function. Unfortunately, the exponential complexity of the full CI wave function severely limits the size of the active space that can practically be considered. For a large-active-space CASSCF algorithm, one must abandon CI in favor of an approach with more desirable scaling properties, such as the density matrix renormalization group (DMRG) or variational two-electron reduced-density method (v2RDM) methods. Using state-of-the-art semidefinite programming techniques and density-fitting approximations, we have developed a v2RDM-driven CASSCF procedure that can treat active spaces with 50 electrons in 50 orbitals and the simultaneous optimization of nearly 2000 orbitals. Analytic gradients are also available for the approach, and v2RDM-CASSCF-optimized bond lengths typically agree with those from configuration-based approaches to within a few hundredths of an angstrom. Lastly, excited electronic states can also be obtained within the v2RDM-CASSCF framework using an extended random phase approximation. However, a proper treatment of excitations from degenerate ground states requires the consideration of pure-state N-representability conditions

Variational Determination of the Two-Electron Reduced Density Matrix: A Tutorial Review

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