We use density-matrix renormalization group, applied to a one-dimensional model of continuum Hamiltonians, to accurately solve chains of hydrogen atoms of various separations and numbers of atoms. We train and test a machine-learned approximation to F [n], the universal part of the electronic density functional, to within quantum chemical accuracy. Our calculation (a) bypasses the standard Kohn-Sham approach, avoiding the need to find orbitals, (b) includes the strong correlation of highly-stretched bonds without any specific difficulty (unlike all standard DFT approximations) and (c) is so accurate that it can be used to find the energy in the thermodynamic limit to quantum chemical accuracy.

As a proof of principle, self-consistent Kohn-Sham calculations are performed with the exact exchange-correlation functional. Finding the exact functional for even one trial density requires solving the interacting Schrödinger equation many times. The density matrix renormalization group method makes this possible for one-dimensional, real-space systems of more than two interacting electrons. We illustrate and explore the convergence properties of the exact KS scheme for both weakly and strongly correlated systems. We also explore the spin-dependent generalization and densities for which the functional is ill defined. Example of a reasonable density (with finite kinetic energy) whose Kohn-Sham potential exists on any grid, but is unphysical in the continuum limit.

Historical methods of functional development in density functional theory have often been guided by analytic conditions that constrain the exact functional one is trying to approximate. Recently, machine-learned functionals have been created by interpolating the results from a small number of exactly solved systems to unsolved systems that are similar in nature. For a simple one-dimensional system, using an exact condition, we find improvements in the learning curves of a machine learning approximation to the non-interacting kinetic energy functional. We also find that the significance of the improvement depends on the nature of the interpolation manifold of the machine-learned functional.

We demonstrate two effects that occur in all diffusive superconducting-magnetic heterostructures with rotating magnetization: the reappearance of singlet |0, 0 correlations deep in the magnetic material and a cascade of s = 1, m = 0, ±1 components (in the two spin−1/2 basis |s, m ). We do so by examining the order parameter and Josephson current through a multilayer with five mutually perpendicular ferromagnets. The properties of the middle layer determine whether the current is due to m = 0 or m = 0 contributions. We conclude that so-called long-and short-range components are present across a proximity system with rotating magnetization.

The order parameter of superconducting pairs penetrating an inhomogeneous magnetic material can acquire a long range triplet component (LRTC) with nonzero spin projection. This state has been predicted and generated in proximity systems and Josephson junctions. We show, using a realistic domain wall of an exchange spring bilayer, how the LRTC emerges and can be tuned with the twisting of the magnetization. We also introduce a new kind of Josephson current reversal, the singlet-LRTC 0-π transition, that can be observed in one and the same system either by tuning the domain wall or by varying temperature.

Successive Linear Programming (SLP) has been used extensively in the refining and petrochemical industries for over 20 years. This paper concentrates on some recent work at Exxon to unify the treatment of nonlinear terms in "mostly linear" models. We first discuss the source of nonlinearities in refining and petrochemical problems and propose a multiplicative formulation for the linearized subproblems to be solved by SLP. We then describe a SLP algorithm which is shown to be related to the concept of trust regions. Finally, we present an example formulation and computational results for a series of large industrial applications.programming: nonlinear, applications, algorithms

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An exponential interaction is constructed so that one-dimensional atoms and chains of atoms mimic the general behavior of their three-dimensional counterparts. Relative to the more commonly used soft-Coulomb interaction, the exponential greatly diminishes the computational time needed for calculating highly accurate quantities with the density matrix renormalization group. This is due to the use of a small matrix product operator and to exponentially vanishing tails. Furthermore, its more rapid decay closely mimics the screened Coulomb interaction in three dimensions. Choosing parameters to best match earlier calculations, we report results for the one dimensional hydrogen atom, uniform gas, and small atoms and molecules both exactly and in the local density approximation.